#6741411

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fgi,e^2=fgi,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, 6648607);