#501987

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28245	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

527 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,24)(13,30)(15,21)(16,19)(17,20)(18,28)(22,34)(23,31)(25,39)(26,37)(27,36)(29,40)(32,38)(33,35);(1,3,4,7)(2,5,6,8)(9,11,17,21,15,20,14,10)(12,25,34,40,28,35,27,37)(13,23,32,38,31,30,16,19)(18,29,22,39,24,26,36,33);(2,6)(5,8)(9,12)(10,18)(11,22)(13,29)(14,27)(15,28)(16,33)(17,34)(19,25)(20,36)(21,24)(23,40)(26,31)(30,37)(32,39)(35,38);(9,13,17,32,15,31,14,16)(10,19,11,23,21,38,20,30)(12,26,34,33,28,29,27,39)(18,35,22,37,24,25,36,40)

Polycyclic presentation:

2^9|b^2=ef,d^2=ei,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409183);