#503976

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

600 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)(9,10,17,20)(11,21,25,34)(12,14,27,16)(13,26,29,40)(15,19,28,31)(18,32,35,24)(22,38,39,30)(23,37,36,33);(2,5)(6,7)(9,11,14,24,17,25,16,32)(10,18,12,21,20,35,27,34)(13,23,15,30,29,36,28,38)(19,33,26,22,31,37,40,39);(9,12,14,20,17,27,16,10)(11,22,24,37,25,39,32,33)(13,26,15,31,29,40,28,19)(18,30,21,36,35,38,34,23);(1,3,4,8)(2,6,5,7)(9,13,16,28,17,29,14,15)(10,19,27,40,20,31,12,26)(11,23,32,38,25,36,24,30)(18,33,34,39,35,37,21,22)

Polycyclic presentation:

2^9|a^2=i,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, 411172);