#72232612

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 8979 conjugacy classes of subgroups; only the 3057 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,c^2=h,e^2=g,c^a=cgh,c^b=chi,d^a=dij,d^c=dij,e^c=eg,e^d=eij,f^b=fg,f^c=fhj,f^d=fj,f^e=fhj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232676
maximal normal subgroup of	#72232676

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