#509321

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

602 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)(15,20)(16,27)(17,21)(18,24)(19,34)(22,29)(23,31)(25,41)(26,36)(28,40)(30,43)(32,35)(33,37)(38,44)(39,42);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,19)(14,20,24,34)(16,28,30,25)(17,26,33,31)(21,36,37,23)(22,38,39,35)(27,40,43,41)(29,44,42,32);(2,4)(5,8,9,10)(6,7,11,12)(14,21)(15,19)(17,31)(20,23)(22,27)(24,37)(25,28)(26,33)(29,32)(34,36)(35,40)(38,41)(39,43)(42,44);(6,11)(7,12)(13,16,18,30)(14,22,24,39)(15,25,19,28)(17,32,33,44)(20,35,34,38)(21,27,37,43)(23,40,36,41)(26,42,31,29)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,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, 416517);