#451153

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1585 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358349);