#7596462

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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 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)(11,15)(12,13)(14,16)(17,18,23,28)(19,26,30,24)(20,32,36,47)(21,37,31,40)(22,38,34,27)(25,41,44,35)(29,42,45,48)(33,39,46,43);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,28,26)(20,33,36,46)(21,34,31,22)(25,42,44,48)(27,37,38,40)(29,35,45,41)(32,43,47,39);(10,13)(14,15)(17,20,34,41,23,36,22,35)(18,25,38,47,28,44,27,32)(19,29,21,33,30,45,31,46)(24,39,37,42,26,43,40,48);(9,11,12,16)(10,14,13,15)(17,21,22,19,23,31,34,30)(18,26,27,37,28,24,38,40)(20,33,35,29,36,46,41,45)(25,43,32,42,44,39,47,48);(1,3,4,8)(2,6,5,7)(17,22,23,34)(18,27,28,38)(19,21,30,31)(20,35,36,41)(24,37,26,40)(25,32,44,47)(29,33,45,46)(39,42,43,48)

Polycyclic presentation:

2^9|a^2=i,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=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, 7503658);