#72342347

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 23688 conjugacy classes of subgroups; only the 29389 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,26)(20,24)(21,25)(22,23)(27,29)(28,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(18,23)(19,25)(20,28)(29,30);(9,11)(10,12)(13,16)(14,15)(20,28)(24,31)(27,32)(29,30);(1,3)(2,5)(4,8)(6,7)(9,12,13,14)(10,11,15,16);(2,6)(5,7)(10,15)(12,14)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31);(1,3)(2,7)(4,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,20,22,28)(18,24,23,31)(19,27,25,32)(21,30,26,29);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,25)(19,23)(20,29)(22,26)(24,27)(28,30)(31,32)

Polycyclic presentation:

2^10|d^2=j,f^2=hij,b^a=bi,d^a=dh,d^c=dj,e^a=ei,e^b=ehij,e^d=ej,f^b=fhij,f^c=fhij,f^d=fj,g^b=gj,g^d=gj,g^e=gi,g^f=gij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

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