#6822317

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1297 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)(10,14)(11,12)(17,18,23,29)(19,25,32,40)(20,33,36,26)(21,27,24,30)(22,28,38,44)(31,45,46,39)(34,47,48,41)(35,43,37,42);(9,10)(11,15)(12,16)(13,14)(17,19,23,32)(18,25,29,40)(20,34,36,48)(21,22,24,38)(26,41,33,47)(27,28,30,44)(31,35,46,37)(39,42,45,43);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,24,37,23,36,21,35)(18,26,30,43,29,33,27,42)(19,31,22,34,32,46,38,48)(25,39,28,41,40,45,44,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,23,24)(18,27,29,30)(19,22,32,38)(20,35,36,37)(25,28,40,44)(26,42,33,43)(31,34,46,48)(39,41,45,47);(10,14)(15,16)(17,22,24,32,23,38,21,19)(18,28,30,40,29,44,27,25)(20,34,37,46,36,48,35,31)(26,41,43,45,33,47,42,39)

Polycyclic presentation:

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

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, 6729513);