#72310925

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|c^2=g,d^2=g,e^2=ij,b^a=bf,c^a=cg,c^b=cf,d^b=dh,d^c=dgj,e^a=eij,e^b=ei,e^c=efhj,e^d=efij,f^c=fi,f^d=fj,g^b=gi,g^e=gj,h^a=hj,h^c=hij,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310969, #72311321, #72340301, …
maximal normal subgroup of	#72310969, #72311321, #72340301, …

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