#7602804

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1368 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)(19,30)(20,34)(21,29)(22,28)(23,27)(24,33)(25,32)(26,38)(31,46)(35,45)(36,44)(37,43)(39,42)(40,48)(41,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,33)(18,24,28,30)(20,35,37,48)(21,32,23,39)(25,29,42,27)(26,41,44,46)(31,36,47,38)(34,40,43,45);(10,14)(11,16)(17,19,23,32,22,33,21,39)(18,25,29,30,28,42,27,24)(20,31,38,35,37,47,36,48)(26,41,34,45,44,46,43,40);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,23,38,22,37,21,36)(18,26,29,34,28,44,27,43)(19,31,32,35,33,47,39,48)(24,40,25,41,30,45,42,46);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,32,33,39)(20,36,37,38)(24,25,30,42)(26,43,44,34)(31,35,47,48)(40,41,45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=fgi,e^2=i,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, 7510000);