#6702105

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,24)(19,27,30,23)(20,32,34,25)(22,26,36,40)(28,38,31,43)(29,39,47,45)(33,41,46,48)(35,44,37,42);(9,11,14,10)(12,15,13,16)(17,19,21,30)(18,23,24,27)(20,33,34,46)(22,28,36,31)(25,41,32,48)(26,43,40,38)(29,35,47,37)(39,42,45,44);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(18,24)(19,28)(22,36)(23,38)(25,32)(27,43)(29,46)(30,31)(33,47)(35,37)(39,41)(45,48);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(18,24)(20,34)(23,27)(26,40)(29,47)(33,46)(35,37)(38,43);(17,20,21,34)(18,25,24,32)(19,29,30,47)(22,37,36,35)(23,39,27,45)(26,44,40,42)(28,46,31,33)(38,41,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fg,e^2=i,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, 6609301);