#6741618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,40)(30,32)(31,44)(33,41)(34,42)(35,39)(36,38)(37,43)(45,48)(46,47);(1,3)(2,5)(4,8)(6,7)(17,19,23,33)(18,24,27,41)(20,34,37,47)(21,30,28,44)(22,31,26,32)(25,42,43,46)(29,39,45,36)(35,48,38,40);(2,6)(5,7)(10,14)(15,16)(17,20,23,37)(18,25,27,43)(19,29,33,45)(21,38,28,35)(22,39,26,36)(24,40,41,48)(30,46,44,42)(31,34,32,47);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,28)(18,26,27,22)(19,30,33,44)(20,35,37,38)(24,32,41,31)(25,39,43,36)(29,42,45,46)(34,48,47,40);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,27)(19,31)(22,26)(24,30)(25,43)(29,34)(32,33)(36,39)(40,46)(41,44)(42,48)(45,47)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648814);