#72300709

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄

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 9363 conjugacy classes of subgroups; only the 3001 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:

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

Polycyclic presentation:

2^10|b^2=i,c^2=i,d^2=h,b^a=bj,c^a=ci,d^a=dh,d^b=dg,d^c=dij,e^a=ei,e^b=ei,e^c=ei,e^d=ei,f^a=fh,f^b=fghi,f^c=fi,f^e=fghi

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