#509095

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

723 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)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,18,33)(16,27,29,43)(17,22,32,40)(21,36,38,28)(23,26,39,31)(25,42,34,41)(30,35,44,37);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,24,33)(16,25,29,34)(17,26,32,31)(21,35,38,37)(22,39,40,23)(27,42,43,41)(28,30,36,44);(2,4)(13,16,19,29)(14,21,24,38)(15,25,18,34)(17,30,32,44)(20,35,33,37)(22,27,40,43)(23,41,39,42)(26,36,31,28);(6,10)(7,12)(14,22)(15,18)(17,31)(20,23)(21,27)(24,40)(25,34)(26,32)(28,30)(33,39)(35,41)(36,44)(37,42)(38,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 416291);