#502865

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60718	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

851 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410061);