#6832163

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 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)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,35)(21,27)(22,28)(23,29)(24,30)(26,36)(31,47)(32,42)(33,43)(34,44)(37,46)(38,45)(39,41)(40,48);(1,3)(2,5)(4,8)(6,7)(17,19,23,34)(18,25,29,44)(20,22,36,38)(21,32,24,40)(26,45,35,28)(27,42,30,48)(31,37,39,33)(41,43,47,46);(10,13)(14,16)(17,20,21,37,23,36,24,33)(18,26,27,43,29,35,30,46)(19,31,40,38,34,39,32,22)(25,41,48,28,44,47,42,45);(2,6)(5,7)(17,21,23,24)(18,27,29,30)(19,32,34,40)(20,33,36,37)(22,31,38,39)(25,42,44,48)(26,46,35,43)(28,47,45,41);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,21,31,23,38,24,39)(18,28,27,47,29,45,30,41)(19,33,40,20,34,37,32,36)(25,43,48,35,44,46,42,26)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6739359);