#403860

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1138 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=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, 311056);