#501639

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	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

723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=ei,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, 408835);