#72328199

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

24505 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|c^2=j,e^2=hij,b^a=bh,c^b=cg,d^c=di,e^b=ei,e^c=ehi,e^d=ei,f^a=fj,f^c=fj,f^e=fghj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328255
maximal normal subgroup of	#72328255

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