#499533

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

798 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406729);