#6761301

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1293 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)(3,6)(4,8)(5,7)(17,18,23,25,21,27,22,28)(19,29,37,41,31,24,32,39)(20,33,38,44,35,26,36,42)(30,40,46,47,34,43,45,48);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,31)(18,24,27,29)(20,34,35,30)(22,32,23,37)(25,39,28,41)(26,43,33,40)(36,46,38,45)(42,48,44,47);(9,11,12,16)(10,13,14,15)(18,25)(22,23)(24,39)(26,42)(27,28)(29,41)(32,37)(33,44)(36,38)(40,47)(43,48)(45,46);(1,4,6,5)(2,7,3,8)(17,20,21,35)(18,26,27,33)(19,30,31,34)(22,36,23,38)(24,40,29,43)(25,42,28,44)(32,45,37,46)(39,47,41,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(19,31)(20,35)(24,29)(26,33)(32,37)(36,38)(39,41)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6668497);