#6813662

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,28)(25,29)(27,38)(30,45)(31,40)(33,41)(34,42)(35,48)(37,44)(39,46)(43,47);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,33)(21,29)(22,34)(23,31)(25,26)(27,44)(28,40)(30,35)(32,41)(36,42)(37,38)(39,43)(45,48)(46,47);(2,6)(5,8)(11,16)(13,15)(17,20,33,19,23,29,21,31)(18,25,41,40,28,32,26,24)(22,37,47,46,38,34,35,30)(27,44,48,39,36,42,43,45);(10,14)(13,15)(17,21,23,33)(18,26,28,41)(19,29,31,20)(22,35,38,47)(24,25,40,32)(27,43,36,48)(30,37,46,34)(39,42,45,44);(17,22,33,47,23,38,21,35)(18,27,41,48,28,36,26,43)(19,30,29,37,31,46,20,34)(24,39,25,42,40,45,32,44)

Polycyclic presentation:

2^9|b^2=gi,c^2=fhi,d^2=i,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, 6720858);