#146511

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#106
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)(5,6,7,8)(9,10,12,17,15,19,14,30)(11,20,23,34,24,38,21,37)(13,26,25,35,29,36,28,18)(16,27,32,22,33,39,31,40);(9,11)(10,16,19,33)(12,21)(13,27)(14,23)(15,24)(17,31,30,32)(18,34,35,37)(20,36,38,26)(22,28)(25,40)(29,39);(2,4)(9,12,15,14)(10,17,19,30)(11,21,24,23)(13,25,29,28)(16,31,33,32)(18,26,35,36)(20,37,38,34)(22,27,40,39)

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,g^a=gi,h^a=hi,d^b=dh,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, 53707);