#72232696

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 9014 conjugacy classes of subgroups; only the 3101 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,20)(19,23,24,22)(26,30)(28,31);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,16)(13,14)(17,18,21,20)(19,23,24,22)(25,26,29,30)(27,28,32,31);(9,12)(10,11)(13,16)(14,15)(17,18,21,20)(19,22,24,23)(25,27)(26,31)(28,30)(29,32);(10,16)(11,13)(25,27)(26,28)(29,32)(30,31);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,32)(30,31);(1,4,5,3)(2,6,7,8)(9,13,14,11)(10,12,16,15)(17,20,21,18)(19,22,24,23)(25,28)(26,32)(27,30)(29,31)

Polycyclic presentation:

2^10|a^2=g,b^2=h,c^2=g,f^2=hj,b^a=bgh,c^a=cgi,c^b=cj,d^a=dgi,d^b=dgi,e^a=ei,e^b=ei,f^a=fgij,f^b=fgi,f^d=fj,f^e=fh

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710, #72328746
maximal normal subgroup of	#72232710, #72328746
maximal quotient of	#72331199

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