#509673

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	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

601 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=i,d^2=e,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=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, 416869);