#503975

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
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	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 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)(12,26)(13,24)(14,20)(16,21)(17,22)(18,30)(19,33)(23,32)(25,34)(27,28)(29,37)(31,38)(35,40)(36,39);(2,5)(6,7)(9,11,15,10,17,20,14,22)(12,27,30,36,32,38,29,40)(13,24,16,19,34,25,33,21)(18,28,26,35,37,31,23,39);(9,12,15,30,17,32,14,29)(10,18,20,26,22,37,11,23)(13,28,16,35,34,31,33,39)(19,27,25,36,21,38,24,40);(1,3,4,8)(2,6,5,7)(9,13,15,16,17,34,14,33)(10,19,20,25,22,21,11,24)(12,28,30,35,32,31,29,39)(18,27,26,36,37,38,23,40)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=ei,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,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, 411171);