#7602803

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(9,10,12,14)(11,15,16,13)(17,18,24,30)(19,25,33,27)(20,34,36,45)(21,39,32,41)(22,29,23,28)(26,38,44,37)(31,42,46,48)(35,40,47,43);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,30,27)(20,35,36,47)(21,22,32,23)(26,42,44,48)(28,39,29,41)(31,38,46,37)(34,40,45,43);(10,14)(11,16)(17,20,23,38,24,36,22,37)(18,26,29,45,30,44,28,34)(19,31,32,47,33,46,21,35)(25,40,39,48,27,43,41,42);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,19,24,32,22,33)(18,27,29,41,30,25,28,39)(20,35,38,31,36,47,37,46)(26,43,45,42,44,40,34,48);(2,5)(6,8)(10,14)(13,15)(20,36)(26,44)(31,46)(34,45)(35,47)(37,38)(40,43)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=fgi,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, 7509999);