#6859384

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

1454 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)(19,24)(20,30)(21,26)(22,36)(23,27)(25,28)(29,38)(31,40)(32,43)(33,37)(34,41)(35,46)(39,42)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,29,34,35)(25,30,40,37)(26,38,41,46)(32,44,45,42)(39,43,48,47);(10,14)(15,16)(17,20,23,33)(18,25,27,40)(19,28,22,31)(21,32,34,45)(24,37,36,30)(26,39,41,48)(29,42,35,44)(38,47,46,43);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,29)(20,32)(22,35)(23,34)(24,38)(25,39)(27,41)(28,42)(30,43)(31,44)(33,45)(36,46)(37,47)(40,48);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,33)(21,34)(25,40)(26,41)(28,31)(29,35)(30,37)(38,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6766580);