#506474

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

835 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)(13,14,16,20,28,35,42,40,17,21,29,36,18,22,30,37)(15,23,25,38,31,44,43,32,24,39,41,33,26,34,27,19);(1,2,3,4)(5,7,8,12)(6,9,10,11)(15,24)(19,32)(23,39)(25,41)(26,31)(27,43)(33,38)(34,44);(2,4)(13,15,17,24)(14,19,21,32)(16,27,29,43)(18,31,28,26)(20,34,36,44)(22,38,35,33)(23,40,39,37)(25,42,41,30);(6,10)(9,11)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413670);