#6832595

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

1315 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,26)(22,37)(23,28)(27,30)(29,38)(31,39)(32,40)(33,41)(34,42)(35,44)(36,43)(45,47)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,28,39)(20,32,36,35)(21,30,33,22)(25,40,43,44)(26,27,41,37)(29,45,46,34)(38,47,48,42);(2,5)(6,8)(10,14)(11,16)(17,20,33,45,23,36,21,34)(18,25,41,47,28,43,26,42)(19,29,30,35,31,46,22,32)(24,38,27,44,39,48,37,40);(10,14)(13,15)(17,21,23,33)(18,26,28,41)(19,30,31,22)(20,34,36,45)(24,27,39,37)(25,42,43,47)(29,35,46,32)(38,44,48,40);(1,3,4,7)(2,6,5,8)(17,22,21,31,23,30,33,19)(18,27,26,24,28,37,41,39)(20,35,34,29,36,32,45,46)(25,40,42,48,43,44,47,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,d^2=i,e^2=gh,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, 6739791);