#6868342

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1465 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)(17,18,19,28,22,27,23,24)(20,31,32,25,34,37,29,40)(21,26,30,41,36,42,35,38)(33,44,45,39,46,47,43,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,35,36,30)(25,31,40,37)(26,41,42,38)(33,43,46,45)(39,47,48,44);(10,14)(15,16)(17,20,22,34)(18,25,27,40)(19,29,23,32)(21,33,36,46)(24,37,28,31)(26,39,42,48)(30,43,35,45)(38,47,41,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,36)(18,26,27,42)(19,30,23,35)(20,33,34,46)(24,38,28,41)(25,39,40,48)(29,43,32,45)(31,44,37,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(18,27)(21,36)(24,28)(25,40)(30,35)(31,37)(33,46)(43,45)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,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, 6775538);