#6820679

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

1558 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)(9,10,13,15)(11,14,12,16)(17,18,21,25)(19,27,30,23)(20,31,34,24)(22,26,35,40)(28,41,43,36)(29,37,33,39)(32,45,44,38)(42,47,46,48);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(19,28)(20,32)(22,35)(23,36)(24,38)(26,40)(27,41)(29,33)(30,43)(31,45)(34,44)(37,39);(9,12,13,11)(10,14,15,16)(17,19,22,28,21,30,35,43)(18,23,26,36,25,27,40,41)(20,33,32,46,34,29,44,42)(24,39,38,48,31,37,45,47);(2,5)(6,8)(20,34)(24,31)(29,33)(32,44)(37,39)(38,45)(42,46)(47,48);(1,3,4,7)(2,6,5,8)(17,20,22,32,21,34,35,44)(18,24,26,38,25,31,40,45)(19,29,28,42,30,33,43,46)(23,37,36,47,27,39,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6727875);