#72312548

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

49639 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(18,23)(19,25)(20,28)(30,32);(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,29)(21,25)(22,23)(24,28)(27,30)(31,32);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,16,15)(18,23)(19,25)(20,28)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16);(2,6)(5,7)(10,13)(14,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,30)(22,28)(23,29)(25,31)(26,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,21)(18,25)(19,23)(20,30)(22,26)(24,31)(27,29)(28,32)

Polycyclic presentation:

2^10|c^2=hi,b^a=bh,c^a=chi,c^b=ci,d^c=dj,e^a=ei,e^b=ei,e^c=ehj,e^d=ehi,f^a=fi,f^b=fi,f^c=fij,f^d=fj,g^c=gj,g^e=gi,g^f=gj

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312560, #72312586, #72342349
maximal normal subgroup of	#72312560, #72312586, #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, 3, F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);