#7593111

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

1348 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=gi,d^2=g,e^2=fi,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,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, 7500307);