#500101

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
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	#60458	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1017 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,18)(14,20)(17,25)(19,26)(21,34)(22,31)(23,29)(24,30)(27,33)(28,35)(32,38)(36,40)(37,39);(2,5)(6,8)(9,11)(10,15)(12,22)(13,21)(14,24)(16,31)(17,28)(18,34)(19,29)(20,30)(23,26)(25,35)(27,36)(32,39)(33,40)(37,38);(1,3,4,7)(2,6,5,8)(10,16)(11,21)(12,25)(13,26)(15,28)(17,19)(18,32)(20,33)(22,36)(23,24)(27,38)(29,40)(30,39)(31,34);(2,5)(6,8)

Polycyclic presentation:

2^9|c^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,f^c=fh,g^c=gi,h^c=hi

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, 407297);