#6818216

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,24,31,40)(20,25,30,39)(21,34,36,26)(22,27,32,41)(29,45,47,38)(33,43,35,42)(37,48,46,44);(2,5)(3,7)(17,19,23,31)(18,24,28,40)(20,32,30,22)(21,35,36,33)(25,41,39,27)(26,43,34,42)(29,46,47,37)(38,48,45,44);(9,11,12,16)(10,13,14,15)(17,20,22,19,23,30,32,31)(18,25,27,24,28,39,41,40)(21,35,37,47,36,33,46,29)(26,43,44,45,34,42,48,38);(2,5)(6,8)(21,36)(26,34)(29,47)(33,35)(37,46)(38,45)(42,43)(44,48);(1,3,4,7)(2,6,5,8)(17,21,22,37,23,36,32,46)(18,26,27,44,28,34,41,48)(19,29,30,35,31,47,20,33)(24,38,39,43,40,45,25,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,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, 6725412);