#72234005

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 9368 conjugacy classes of subgroups; only the 3061 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)(4,7)(5,6)(10,11)(14,15)(25,26,29,32)(27,30,31,28);(1,2)(3,8)(4,7)(5,6)(25,27)(26,30)(28,32)(29,31);(2,6)(3,7)(9,10)(11,13)(12,15)(14,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,31)(27,32)(29,30);(10,11)(14,15)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,31)(29,32);(1,3)(2,4)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,23)(22,24)(25,26)(27,28)(29,32)(30,31);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,24)(21,22)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^10|a^2=g,e^2=i,c^a=ch,c^b=cghi,d^a=dg,d^c=dhj,e^a=eh,e^b=ehi,e^c=eh,e^d=ei,f^c=fhj,f^e=fj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234056
maximal normal subgroup of	#72234056

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