#72249545

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

54649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249567, #72262295, #72262300, …
maximal normal subgroup of	#72249567, #72262295, #72262300, …

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