#7594156

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

1437 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=fgi,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7501352);