#7595097

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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(12,13)(14,16)(17,18)(19,28)(20,31)(21,36)(22,26)(23,39)(24,30)(25,35)(27,38)(29,41)(32,40)(33,44)(34,47)(37,43)(42,48)(45,46);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,32,34,33)(21,23,37,38)(25,41,42,46)(27,43,39,36)(29,35,45,48)(31,44,47,40);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,23,35,22,34,38,48)(18,25,27,31,26,42,39,47)(19,29,21,33,30,45,37,32)(24,40,36,46,28,44,43,41);(9,11,12,16)(10,14,13,15)(17,21,23,30,22,37,38,19)(18,24,27,36,26,28,39,43)(20,33,35,45,34,32,48,29)(25,40,31,46,42,44,47,41);(2,5)(6,8)(18,26)(20,34)(24,28)(27,39)(29,45)(32,33)(35,48)(36,43)

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=g,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, 7502293);