#7603911

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

1375 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,16,12,15)(17,18,23,27)(19,28,31,24)(20,32,37,48)(21,38,30,41)(22,39,34,26)(25,42,45,36)(29,46,47,43)(33,44,35,40);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,31)(18,24,27,28)(20,33,37,35)(21,22,30,34)(25,43,45,46)(26,41,39,38)(29,42,47,36)(32,40,48,44);(10,14)(15,16)(17,20,34,42,23,37,22,36)(18,25,39,48,27,45,26,32)(19,29,30,35,31,47,21,33)(24,40,41,46,28,44,38,43);(9,11,13,12)(10,15,14,16)(17,21,22,31,23,30,34,19)(18,24,26,38,27,28,39,41)(20,35,36,29,37,33,42,47)(25,44,32,46,45,40,48,43);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(17,22,23,34)(18,26,27,39)(19,30,31,21)(20,36,37,42)(24,41,28,38)(25,32,45,48)(29,35,47,33)(40,46,44,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,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=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, 7511107);