#509889

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

524 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)(7,8)(9,11)(10,12)(13,14,19,31,30,39,29,21,18,25,16,23,17,24,33,22)(15,20,32,42,41,44,40,34,28,38,26,36,27,37,43,35);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,28)(14,20,25,38)(16,26,19,32)(17,27,30,41)(21,34,22,35)(23,36,31,42)(24,37,39,44)(29,40,33,43);(2,4)(5,8,9,10)(6,7,11,12)(14,21)(16,29)(17,30)(19,33)(20,34)(22,25)(23,24)(26,40)(27,41)(31,39)(32,43)(35,38)(36,37)(42,44);(6,11)(7,12)(14,22)(16,29)(17,30)(19,33)(20,35)(21,25)(23,39)(24,31)(26,40)(27,41)(32,43)(34,38)(36,44)(37,42)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417085);