#7594688

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=h,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7501884);