#403822

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
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)(9,10,16,19,15,12,14,20)(11,21,31,25,24,33,23,17)(13,27,28,26,30,34,29,18)(22,32,36,40,38,39,37,35);(1,2,4,6)(3,8,7,5)(9,11,16,31,15,24,14,23)(10,17,20,33,12,25,19,21)(13,22,29,37,30,38,28,36)(18,32,27,40,26,39,34,35);(1,3,4,7)(2,5,6,8)(9,12,15,10)(11,17,24,25)(13,26,30,18)(14,20,16,19)(21,31,33,23)(22,32,38,39)(27,29,34,28)(35,37,40,36);(9,13,14,28,15,30,16,29)(10,18,19,34,12,26,20,27)(11,22,23,36,24,38,31,37)(17,32,21,35,25,39,33,40)

Polycyclic presentation:

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

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, 311018);