#6868102

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775298);