#72300730

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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 9464 conjugacy classes of subgroups; only the 3005 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=ij,d^2=hi,e^2=ij,b^a=bi,c^b=cij,d^b=dg,d^c=dh,e^c=eij,e^d=ehj,f^a=fhi,f^b=fghj,f^c=fij,f^e=fghj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300808
maximal normal subgroup of	#72300808

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