#403740

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=gi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^d=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, 310936);