#503978

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	#28504	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=e,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,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, 411174);