#505734

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

539 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)(11,13)(12,14)(15,16)(17,18,24,30,21,20,38,41,22,28,23,29,25,27,42,40)(19,31,36,44,33,43,47,45,34,26,35,32,37,39,46,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,23,35,21,33,42,46,22,34,24,36,25,37,38,47)(18,26,29,44,20,39,40,45,28,31,30,32,27,43,41,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,22,27)(18,25,28,21)(19,32,34,44)(23,29,24,30)(26,35,31,36)(33,45,37,48)(38,40,42,41)(39,47,43,46);(2,5)(6,8)(17,21,22,25)(18,27,28,20)(19,33,34,37)(23,42,24,38)(26,43,31,39)(29,41,30,40)(32,45,44,48)(35,46,36,47)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412930);