#6909312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh

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, 6816508);