#72340574

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
prime radical	#7667013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

45743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=j,c^2=hj,d^2=hij,g^2=j,c^a=ch,c^b=ci,d^a=dhi,d^b=dh,d^c=dj,e^a=eh,e^c=ehij,e^d=eh,f^a=fi,f^c=fi,f^d=fhij,g^b=gj

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340595
maximal subgroup of	#72340582, #72340595
maximal normal subgroup of	#72340582, #72340595

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