#6741408

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,22)(19,24,30,29)(20,25,33,36)(21,26,34,35)(27,37,41,31)(28,38,42,43)(32,39,45,46)(40,47,48,44);(1,3)(2,5)(4,8)(6,7)(17,19,23,30)(18,24,22,29)(20,31,33,37)(21,28,34,42)(25,27,36,41)(26,38,35,43)(32,44,45,47)(39,40,46,48);(2,6)(5,7)(10,14)(15,16)(17,20,23,33)(18,25,22,36)(19,27,30,41)(21,32,34,45)(24,37,29,31)(26,39,35,46)(28,40,42,48)(38,47,43,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,28)(20,32)(22,35)(23,34)(24,38)(25,39)(27,40)(29,43)(30,42)(31,44)(33,45)(36,46)(37,47)(41,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,22)(19,29)(21,34)(24,30)(25,36)(27,31)(28,38)(32,45)(37,41)(40,47)(42,43)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6648604);