#403861

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×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1178 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)(9,10,15,18,14,20,13,19)(11,21,29,22,25,32,24,16)(12,26,30,34,28,17,27,33)(23,37,40,31,39,35,38,36);(1,2,4,6)(3,8,7,5)(9,11,13,24,14,25,15,29)(10,16,18,21,20,22,19,32)(12,23,27,38,28,39,30,40)(17,31,33,35,26,36,34,37);(1,3,4,7)(2,5,6,8)(9,10,14,20)(11,22,25,16)(12,17,28,26)(13,18,15,19)(21,24,32,29)(23,36,39,31)(27,33,30,34)(35,38,37,40);(9,12,13,27,14,28,15,30)(10,17,18,33,20,26,19,34)(11,23,24,38,25,39,29,40)(16,31,21,35,22,36,32,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,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, 311057);