#546451

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=h,d^2=fi,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 453647);