#7605764

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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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)(9,10,13,12)(11,14,16,15)(17,18)(19,26)(20,32)(21,37)(22,27)(23,39)(24,30)(25,36)(28,38)(29,45)(31,41)(33,43)(34,40)(35,48)(42,47)(44,46);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,30)(18,24,27,26)(20,33,35,34)(21,38,31,23)(25,42,44,45)(28,37,39,41)(29,46,47,36)(32,40,48,43);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,23,36,22,35,38,46)(18,25,28,32,27,44,39,48)(19,29,31,33,30,47,21,34)(24,40,41,42,26,43,37,45);(1,3,5,4)(2,7,8,6)(11,16)(14,15)(17,21,23,19,22,31,38,30)(18,26,28,37,27,24,39,41)(20,34,36,29,35,33,46,47)(25,43,32,45,44,40,48,42);(1,4,5,3)(2,7,8,6)(18,27)(20,35)(24,26)(28,39)(29,47)(33,34)(36,46)(37,41)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fhi,d^2=fgi,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,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, 7512960);