#7079433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

1438 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)(17,18,22,26,23,27,36,42)(19,28,30,38,31,41,21,24)(20,32,35,48,34,25,39,40)(29,44,46,37,45,43,33,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,27,38)(20,29,34,45)(21,36,30,22)(25,37,32,47)(26,41,42,28)(33,39,46,35)(40,44,48,43);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,40)(27,32)(28,43)(30,46)(31,45)(36,39)(38,47)(41,44)(42,48);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,30,36,31)(18,24,26,28,27,38,42,41)(20,33,35,29,34,46,39,45)(25,37,40,43,32,47,48,44);(10,14)(13,15)(20,34)(25,32)(29,45)(33,46)(35,39)(37,47)(40,48)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=fgi,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

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, 6986629);