#7595353

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	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,30)(20,34)(21,39)(22,27)(23,29)(24,28)(25,32)(26,37)(31,46)(33,41)(35,45)(36,44)(38,43)(40,48)(42,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,25,27,30)(20,35,36,48)(21,23,33,24)(26,42,43,46)(28,41,29,39)(31,38,47,37)(34,40,44,45);(10,14)(11,16)(17,20,23,37,22,36,24,38)(18,26,28,34,27,43,29,44)(19,31,21,35,32,47,33,48)(25,40,39,42,30,45,41,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,19,22,33,23,32)(18,25,29,41,27,30,28,39)(20,35,38,31,36,48,37,47)(26,40,44,46,43,45,34,42);(2,5)(6,8)(10,14)(13,15)(18,27)(20,36)(25,30)(28,29)(31,47)(35,48)(37,38)(39,41)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=fg,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^c=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, 7502549);