#508990

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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)(19,27)(23,36)(28,43)(29,44)(30,35)(33,47)(37,42)(40,48);(2,5)(6,8)(17,18,21,25,32,39,46,45,20,24,31,38,22,26,34,41)(19,28,29,42,35,48,47,36,27,43,44,37,30,40,33,23);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,20,27)(18,23,24,36)(21,33,31,47)(22,35,32,30)(25,40,38,48)(26,42,39,37)(28,45,43,41)(29,46,44,34);(3,7)(6,8)(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416186);