#7592779

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1286 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)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,32)(21,36)(22,38)(23,27)(24,31)(25,34)(26,39)(29,42)(30,41)(33,40)(35,46)(37,43)(44,47)(45,48);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,27,28)(20,33,35,37)(21,22,30,39)(25,42,44,48)(26,41,38,36)(29,34,45,47)(32,43,46,40);(17,20,22,34,23,35,39,47)(18,25,26,32,27,44,38,46)(19,29,21,37,31,45,30,33)(24,40,36,48,28,43,41,42);(11,15)(13,16)(17,21,22,31,23,30,39,19)(18,24,26,36,27,28,38,41)(20,33,34,29,35,37,47,45)(25,43,32,42,44,40,46,48);(1,3,4,8)(2,6,5,7)(17,22,23,39)(18,26,27,38)(19,30,31,21)(20,34,35,47)(24,41,28,36)(25,32,44,46)(29,33,45,37)(40,42,43,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=g,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=cgi,d^b=dh,g^b=gi,h^b=hi,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, 7499975);