#508999

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,g^c=gh,h^c=hi

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, 416195);