#72268481

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁹×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

9995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 56 points:

(2,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,24)(21,23)(26,31)(27,35)(33,38)(34,45)(36,51)(37,41)(48,50)(53,54);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,22)(20,23)(21,24)(27,35)(32,44)(36,51)(37,41)(46,52)(47,49)(53,54)(55,56);(2,5)(3,7)(10,14)(11,16)(19,24)(20,22)(25,26)(27,32)(28,38)(29,34)(30,31)(33,40)(35,44)(36,52)(37,47)(39,50)(41,49)(42,48)(43,45)(46,51)(53,56)(54,55);(2,5)(6,8)(11,16)(13,15)(18,23)(20,22)(25,27)(26,32)(28,36)(29,41)(30,35)(31,44)(33,46)(34,49)(37,43)(38,52)(39,54)(40,51)(42,53)(45,47)(48,56)(50,55);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,28,30,40)(26,33,31,38)(27,36,35,51)(29,42,43,39)(32,46,44,52)(34,50,45,48)(37,54,41,53)(47,56,49,55);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,29,30,43)(26,34,31,45)(27,37,35,41)(28,39,40,42)(32,47,44,49)(33,48,38,50)(36,53,51,54)(46,55,52,56)

Polycyclic presentation:

2^10|a^2=g,b^2=g,e^2=j,f^2=j,b^a=bg,c^a=cgj,c^b=ch,d^a=dghj,d^b=dgj,e^a=ei,e^b=eh,e^c=eij,e^d=ei,f^a=fh,f^b=fhi,f^c=fi,f^d=fj,f^e=fj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268530, #72268531, #72271511, …
Fitting subgroup of	#72268530
maximal subgroup of	#72268486, #72268500, #72268530
maximal normal subgroup of	#72268486, #72268500, #72268530

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);