#72312174

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 7841 conjugacy classes of subgroups; only the 29401 normal subgroups are shown.

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

Polycyclic presentation:

2^10|b^2=h,c^2=hi,e^2=hij,c^a=chi,c^b=chj,d^b=dh,d^c=dj,e^a=eij,e^b=eij,e^c=eij,e^d=ej,f^a=fij,f^b=fj,f^c=fij,f^e=fj,g^c=gj,g^e=gij,g^f=gj

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312180, #72342349
maximal normal subgroup of	#72312180, #72342349

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