#501986

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	#28247	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)(11,22)(12,28)(13,20)(14,19)(15,21)(16,25)(17,29)(18,31)(23,24)(26,35)(27,37)(30,36)(32,38)(33,40)(34,39);(1,3,4,7)(2,5,6,8)(9,11,13,25,15,27,31,26)(10,16,18,22,21,35,20,37)(12,24,29,39,30,40,14,32)(17,33,28,34,19,23,36,38);(2,6)(5,8)(9,10)(11,23)(12,19)(13,18)(14,28)(15,21)(16,32)(17,30)(20,31)(22,24)(25,38)(26,34)(27,33)(29,36)(35,39)(37,40);(9,12,13,29,15,30,31,14)(10,17,18,28,21,19,20,36)(11,24,25,39,27,40,26,32)(16,33,22,34,35,23,37,38)

Polycyclic presentation:

2^9|b^2=ef,d^2=e,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, 409182);