#509574

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

519 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=ei,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, 416770);