#6868295

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(17,18,23,37,22,27,19,24)(20,30,28,25,33,38,32,41)(21,34,35,39,36,26,29,42)(31,45,43,48,47,44,46,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,24,27,37)(20,28,33,32)(21,35,36,29)(25,38,41,30)(26,42,34,39)(31,46,47,43)(40,44,48,45);(10,14)(15,16)(17,20,22,33)(18,25,27,41)(19,28,23,32)(21,31,36,47)(24,38,37,30)(26,40,34,48)(29,43,35,46)(39,45,42,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,36)(18,26,27,34)(19,29,23,35)(20,31,33,47)(24,39,37,42)(25,40,41,48)(28,43,32,46)(30,44,38,45);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(21,36)(26,34)(29,35)(31,47)(39,42)(40,48)(43,46)(44,45)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6775491);