#72249540

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 6519 conjugacy classes of subgroups; only the 29669 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=h,b^a=bi,c^a=cij,e^c=ej,e^d=eij,f^b=fij,f^e=fj,g^a=gij,g^b=gj,g^e=ghi

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249567, #72256124
maximal normal subgroup of	#72249567, #72256124

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