#72286050

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#51758
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#1082861
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,7,11)(3,8,6,19)(4,14)(5,17,18,10)(9,16)(12,24)(13,27,21,23)(15,26)(20,22)(25,30)(28,32)(29,31);(1,3,7,6)(2,8,11,19)(4,12,16,20)(5,13,18,21)(9,24,14,22)(10,23,17,27)(15,28,30,29)(25,32,26,31);(4,12,16,20)(9,24,14,22)(15,29,30,28)(25,31,26,32);(1,4)(2,9)(3,12)(5,15)(6,20)(7,16)(8,22)(10,26)(11,14)(13,28)(17,25)(18,30)(19,24)(21,29)(23,32)(27,31);(1,5,7,18)(2,10)(3,13,6,21)(4,15)(8,23)(9,25,14,26)(11,17)(12,28)(16,30)(19,27)(20,29)(22,31,24,32)

Polycyclic presentation:

2^10|a^2=h,b^2=j,c^2=hj,e^2=hi,f^2=j,b^a=bhj,c^a=chj,d^a=di,d^b=di,d^c=df,e^a=eg,e^c=ehj,e^d=eghi,f^a=fj,f^d=fj,f^e=fj,g^d=gj,g^e=gj,h^d=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286289
maximal normal subgroup of	#72286289

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