#6825812

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=gi,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, 6733008);