#6842568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,29,23)(20,31,33,24)(22,26,34,38)(28,39,41,35)(30,36,43,40)(32,44,45,37)(42,46,48,47);(9,11,12,16)(10,13,14,15)(19,28)(20,32)(22,34)(23,35)(24,37)(26,38)(27,39)(29,41)(30,43)(31,44)(33,45)(36,40);(2,6)(7,8)(10,14)(11,16)(17,19,22,28,21,29,34,41)(18,23,26,35,25,27,38,39)(20,30,32,42,33,43,45,48)(24,36,37,46,31,40,44,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(19,29)(23,27)(28,41)(30,43)(35,39)(36,40)(42,48)(46,47);(1,4,5,3)(2,8,6,7)(17,20,22,32,21,33,34,45)(18,24,26,37,25,31,38,44)(19,30,28,42,29,43,41,48)(23,36,35,46,27,40,39,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6749764);