#6741280

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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,27)(19,24,32,41)(20,25,36,44)(21,26,37,22)(28,39,45,47)(29,31,40,30)(33,42,46,48)(34,35,43,38);(1,3)(2,5)(4,8)(6,7)(17,19,23,32)(18,24,27,41)(20,33,36,46)(21,29,37,40)(22,30,26,31)(25,42,44,48)(28,38,45,35)(34,47,43,39);(2,6)(5,7)(10,14)(15,16)(17,20,23,36)(18,25,27,44)(19,28,32,45)(21,34,37,43)(22,38,26,35)(24,39,41,47)(29,42,40,48)(30,33,31,46);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,29)(20,34)(22,27)(23,37)(24,31)(25,35)(28,42)(30,41)(32,40)(33,47)(36,43)(38,44)(39,46)(45,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,30)(21,37)(22,26)(24,29)(28,33)(31,32)(34,43)(35,38)(39,42)(40,41)(45,46)(47,48)

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,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, 6648476);