#6826640

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1258 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)(11,15)(12,14)(13,16)(17,18)(19,24)(20,32)(21,35)(22,27)(23,28)(25,31)(26,37)(29,44)(30,40)(33,42)(34,41)(36,47)(38,48)(39,46)(43,45);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,40)(20,33,31,23)(21,36,37,34)(25,28,32,42)(26,41,35,47)(29,45,46,38)(39,48,44,43);(2,5)(6,8)(11,16)(13,15)(17,20,23,19,22,31,33,30)(18,25,28,40,27,32,42,24)(21,34,38,29,37,36,45,46)(26,41,43,44,35,47,48,39);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(21,37)(26,35)(29,46)(34,36)(38,45)(39,44)(41,47)(43,48);(17,21,33,45,22,37,23,38)(18,26,42,48,27,35,28,43)(19,29,20,34,30,46,31,36)(24,39,32,47,40,44,25,41)

Polycyclic presentation:

2^9|b^2=i,c^2=fh,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,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, 6733836);