#72326666

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄

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 9712 conjugacy classes of subgroups; only the 3017 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=j,b^a=bg,c^b=cgh,d^a=dhj,d^c=dhj,e^a=ei,e^c=ei,f^b=fi,f^c=fj,f^d=fi,f^e=fgij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326780
maximal normal subgroup of	#72326780

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