#7605124

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1374 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=fh,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7512320);