#72340357

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233549	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=gh,e^2=fghj,b^a=bh,c^a=cf,c^b=cfghij,d^a=dij,d^c=dij,e^a=eh,e^b=efj,e^c=ehij,e^d=eij,f^b=fi,f^c=fij,f^e=fj,g^a=gij,g^c=gj,g^e=gi,h^b=hij,h^c=hj,h^e=hij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340359, #72340377, #72340378, …
maximal normal subgroup of	#72340359, #72340377, #72340378, …

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