#6651438

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6558634);