#509794

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

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, 416990);