#500463

order	512
nilpotency class	4
isoclinism	isoclinic class of #499757
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

805 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,27)(14,19)(17,25)(18,28)(20,36)(21,31)(22,39)(23,32)(24,30)(26,34)(29,37)(33,40)(35,38);(2,5)(6,7)(9,11)(10,15)(12,24)(13,20)(14,23)(16,30)(17,31)(18,35)(19,32)(21,25)(22,26)(27,36)(28,38)(29,33)(34,39)(37,40);(10,16)(11,20)(12,25)(13,26)(15,29)(17,19)(18,27)(21,22)(23,38)(24,37)(28,40)(30,35)(31,36)(32,39);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407659);