#7590472

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,23,28)(19,26,30,44)(20,25,32,40)(21,35,31,24)(22,38,37,27)(29,43,45,41)(33,39,36,48)(34,47,46,42);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,30,37,21)(18,24,27,26,28,35,38,44)(20,29,34,33,32,45,46,36)(25,39,42,43,40,48,47,41);(1,3,4,8)(2,5,6,7)(17,20,23,32)(18,25,28,40)(19,29,30,45)(21,36,31,33)(22,34,37,46)(24,39,35,48)(26,43,44,41)(27,42,38,47);(2,6)(5,7)(11,15)(13,16)(19,30)(20,32)(21,31)(24,35)(25,40)(26,44)(34,46)(42,47);(17,21,37,30,23,31,22,19)(18,26,38,24,28,44,27,35)(20,33,46,29,32,36,34,45)(25,41,47,48,40,43,42,39)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,e^2=fi,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^c=ei,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, 7497668);