#499498

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

621 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)(7,10)(8,9)(11,12)(13,14,19,30,29,38,28,21,18,22,16,23,17,24,32,37)(15,20,31,41,39,44,40,33,25,36,26,34,27,35,42,43);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,25)(19,31)(21,33)(22,36)(23,34)(24,35)(28,40)(29,39)(30,41)(32,42)(37,43)(38,44);(2,4)(14,21)(16,28)(17,29)(19,32)(20,33)(22,37)(23,24)(26,40)(27,39)(30,38)(31,42)(34,35)(36,43)(41,44);(5,7,8,11)(6,10,9,12)(14,22)(15,25)(21,37)(23,30)(24,38)(26,31)(27,39)(40,42)

Polycyclic presentation:

2^9|a^2=fi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,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, 406694);