#7079463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1434 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,26,36,43)(19,28,31,39,30,44,21,24)(20,25,35,41,34,42,47,48)(29,45,33,40,32,46,37,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,24,26,39)(20,32,34,29)(21,36,31,22)(25,40,42,38)(27,44,43,28)(33,47,37,35)(41,45,48,46);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,25,26,42)(19,29,30,32)(21,37,31,33)(22,35,36,47)(24,38,39,40)(27,41,43,48)(28,45,44,46);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,31,36,30)(18,24,27,28,26,39,43,44)(20,33,35,32,34,37,47,29)(25,40,41,46,42,38,48,45);(10,14)(13,15)(18,26)(19,30)(20,34)(21,31)(27,43)(35,47)(38,40)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6986659);