#550840

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

2110 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)(9,10,14,11)(12,15,20,17)(13,18,21,16)(19,23,24,22)(26,30)(28,36)(31,33)(34,40);(6,8)(10,11)(15,17)(16,18)(22,23)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(5,6)(7,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2,4,3)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3,4,2)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=gi,e^2=fgi,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, 458036);