#509888

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 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)(13,14,19,33,29,38,30,21,16,20,17,23,18,24,34,36)(15,25,42,39,32,44,31,22,26,43,27,41,28,40,35,37);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(14,20)(15,26)(21,36)(23,33)(24,38)(27,42)(28,32)(31,35);(2,4)(5,8,9,10)(6,7,11,12)(14,21)(17,30)(18,29)(19,34)(20,36)(22,25)(23,24)(27,31)(28,32)(33,38)(35,42)(37,43)(39,44)(40,41);(6,11)(7,12)(13,15,16,26)(14,22,20,37)(17,31,19,35)(18,32,29,28)(21,25,36,43)(23,40,33,44)(24,41,38,39)(27,34,42,30)

Polycyclic presentation:

2^9|a^2=fi,c^2=ei,d^2=i,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, 417084);