#6867989

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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)(17,18,23,36,21,26,22,27)(19,28,37,39,30,24,31,43)(20,32,35,25,33,41,34,40)(29,44,47,42,45,48,46,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,30)(18,24,26,28)(20,29,33,45)(22,31,23,37)(25,38,40,42)(27,39,36,43)(32,48,41,44)(34,46,35,47);(10,14)(15,16)(17,20,21,33)(18,25,26,40)(19,29,30,45)(22,35,23,34)(24,38,28,42)(27,41,36,32)(31,47,37,46)(39,44,43,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,26)(19,30)(25,40)(27,36)(29,45)(31,37)(32,41)(46,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(19,30)(24,28)(29,45)(31,37)(38,42)(39,43)(44,48)(46,47)

Polycyclic presentation:

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

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, 6775185);