#509232

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	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

718 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^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, 416428);