#6742292

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,30,37)(20,24,31,38)(21,25,33,40)(27,34,29,36)(28,35,43,47)(32,39,44,48)(41,45,42,46);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,28)(21,27)(22,30)(24,35)(25,34)(26,37)(29,33)(31,43)(32,41)(36,40)(38,47)(39,45)(42,44)(46,48);(10,14)(13,15)(19,27)(21,33)(23,34)(25,40)(28,41)(29,30)(32,44)(35,45)(36,37)(39,48)(42,43)(46,47);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,39)(26,38)(27,41)(29,42)(30,43)(33,44)(34,45)(36,46)(37,47)(40,48);(1,5,6,4)(2,7,3,8)(11,16)(13,15)(19,27)(20,31)(21,33)(23,34)(24,38)(25,40)(28,42)(29,30)(35,46)(36,37)(41,43)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 6649488);