#72334198

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 8780 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) = 128

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

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

Polycyclic presentation:

2^10|c^2=i,e^2=g,b^a=bj,c^a=ci,c^b=cg,d^c=dgh,e^a=ei,e^b=eh,e^c=egi,e^d=egh,f^a=fi,f^b=fhj,f^d=fgj,f^e=fgh

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334244
maximal normal subgroup of	#72334244

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