#7603057

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

1345 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)(19,24)(20,31)(21,35)(22,37)(23,27)(25,34)(26,36)(28,43)(29,42)(30,39)(32,46)(33,47)(38,45)(40,44)(41,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,27,39)(20,32,33,45)(21,36,29,22)(25,40,41,43)(26,42,37,35)(28,34,44,48)(31,46,47,38);(10,14)(11,16)(17,20,22,34,23,33,36,48)(18,25,26,31,27,41,37,47)(19,28,29,45,30,44,21,32)(24,38,35,43,39,46,42,40);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,19,23,29,36,30)(18,24,26,35,27,39,37,42)(20,32,34,28,33,45,48,44)(25,38,31,43,41,46,47,40);(2,5)(6,8)(10,14)(13,15)(20,33)(25,41)(28,44)(31,47)(32,45)(34,48)(38,46)(40,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,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=cgi,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, 7510253);