#72300846

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 8112 conjugacy classes of subgroups; only the 2989 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)(3,6)(4,7)(5,8)(19,23)(21,24)(27,31)(29,32);(1,2,5,8)(3,6,4,7)(9,10,13,12)(11,14,16,15)(19,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(25,26)(27,32)(28,30)(29,31);(3,4)(6,7)(10,12)(14,15)(17,18,20,22)(19,21,23,24)(25,26,28,30)(27,29,31,32);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,18)(19,24)(20,22)(21,23)(25,26)(27,29)(28,30)(31,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,27)(26,29)(28,31)(30,32)

Polycyclic presentation:

2^10|b^2=g,c^2=g,d^2=h,e^2=g,b^a=bi,d^b=dj,d^c=dgj,e^d=egh,f^a=fh,f^b=fghj,f^c=fij,f^e=fghj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300929
maximal normal subgroup of	#72300929

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