#503057

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60693	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^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,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410253);