#72268321

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21746
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂³×Q₈	C₂⁴
prime radical	C₂⁶	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#543 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

2605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,8,5,9,12,16)(3,6,10,14,11,7,13,15)(17,18)(19,23)(20,30,33,24)(21,34,36,25)(22,26)(27,40,31,37)(28,43,35,38)(29,39)(32,41)(42,47)(44,46)(45,48);(2,6,9,7)(4,12)(8,14,16,15)(10,13)(17,19,22,29)(18,23,26,39)(20,31,33,27)(21,35,36,28)(24,40,30,37)(25,38,34,43)(32,42,46,45)(41,48,44,47);(17,20)(18,24)(19,27)(21,32)(22,33)(23,37)(25,41)(26,30)(28,42)(29,31)(34,44)(35,45)(36,46)(38,47)(39,40)(43,48);(17,21)(18,25)(19,28)(20,32)(22,36)(23,38)(24,41)(26,34)(27,42)(29,35)(30,44)(31,45)(33,46)(37,47)(39,43)(40,48)

Polycyclic presentation:

2^10|a^2=f,b^2=hi,e^2=j,f^2=j,b^a=be,c^a=cg,c^b=ci,d^a=dg,d^b=dgi,e^a=eh,e^d=ei,f^b=fhj,f^c=fi,f^d=fi,f^e=fj,g^a=gi,h^a=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268323, #72268331, #72268332
maximal normal subgroup of	#72268323, #72268331, #72268332

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