#509818

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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	#24778	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

719 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)(13,14,20,35,28,39,27,38,19,24,18,23,15,21,36,44)(16,29,32,40,25,43,33,22,17,31,30,37,26,42,34,41);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,28)(14,21,24,39)(16,25,17,26)(18,27,20,36)(22,37,41,40)(23,38,35,44)(29,43,31,42)(30,34,32,33);(2,4)(13,16,19,17)(14,22,24,41)(15,25,28,26)(18,33,20,34)(21,37,39,40)(23,42,35,43)(27,30,36,32)(29,44,31,38);(6,10)(7,12)(13,17,19,16)(14,22,24,41)(15,26,28,25)(18,34,20,33)(21,37,39,40)(23,42,35,43)(27,32,36,30)(29,44,31,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,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, 417014);