#7594660

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

1350 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)(21,24)(22,39)(23,26)(25,35)(27,32)(28,30)(31,43)(34,45)(36,42)(37,44)(38,41)(40,48)(46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,30,38,23,32,22,21)(18,24,39,27,26,41,28,29)(20,34,46,36,35,31,37,48)(25,42,47,45,33,40,44,43);(2,6)(3,7)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,31)(21,36)(22,37)(23,35)(24,40)(26,33)(27,43)(28,44)(29,45)(30,46)(32,34)(38,48)(39,47)(41,42);(2,6)(5,8)(10,14)(13,16)(18,26)(19,32)(20,35)(21,38)(28,39)(37,46)(40,42)(43,45);(17,21,22,32,23,38,30,19)(18,27,28,24,26,29,39,41)(20,36,37,34,35,48,46,31)(25,43,44,40,33,45,47,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=hi,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,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, 7501856);