#6828097

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1324 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)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,27,29,23)(20,24,33,38)(22,26,32,37)(28,39,41,34)(30,40,43,35)(31,36,44,46)(42,47,48,45);(1,3,6,2)(4,8,5,7)(19,28)(20,31)(22,32)(23,34)(24,36)(26,37)(27,39)(29,41)(30,43)(33,44)(35,40)(38,46);(10,13)(14,16)(17,19,22,28,21,29,32,41)(18,23,26,34,25,27,37,39)(20,30,31,42,33,43,44,48)(24,35,36,45,38,40,46,47);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(19,29)(23,27)(28,41)(30,43)(34,39)(35,40)(42,48)(45,47);(1,5,6,4)(2,7,3,8)(17,20,32,44,21,33,22,31)(18,24,37,46,25,38,26,36)(19,30,41,48,29,43,28,42)(23,35,39,47,27,40,34,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6735293);