#503284

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

770 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:

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

Polycyclic presentation:

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

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, 410480);