#72267607

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	C₂⁶	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

15473 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:

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

Polycyclic presentation:

2^10|c^2=h,d^2=h,e^2=j,f^2=j,c^a=cg,c^b=cj,d^a=dj,d^b=dgj,d^c=dh,e^c=ei,f^d=fi,f^e=fj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267608
maximal normal subgroup of	#72267608
maximal quotient of	#72267615

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, 3, F.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);