#146521

order	512
nilpotency class	4
isoclinism	isoclinic class of #11663
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dhi,f^b=fi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53717);