#6828804

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(11,15)(12,14)(13,16)(17,18)(19,30)(20,34)(21,29)(22,38)(23,28)(24,32)(25,33)(26,35)(27,40)(31,41)(36,42)(37,43)(39,44)(45,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,28,30)(20,35,33,21)(22,36,40,39)(25,26,34,29)(27,42,38,44)(31,37,47,48)(41,45,46,43);(2,5)(6,8)(10,14)(11,16)(17,20,21,19,23,33,35,32)(18,25,29,24,28,34,26,30)(22,39,37,47,40,36,48,31)(27,44,45,46,38,42,43,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,35)(18,26,28,29)(19,20,32,33)(22,37,40,48)(24,34,30,25)(27,43,38,45)(31,36,47,39)(41,44,46,42);(17,22,21,37,23,40,35,48)(18,27,29,45,28,38,26,43)(19,31,33,39,32,47,20,36)(24,41,34,44,30,46,25,42)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=fgi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6736000);