#573022

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

2106 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)(10,14)(12,19)(15,20)(18,23)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,2,4,3)(5,6,7,8)(10,14)(11,17)(16,21)(18,23)(26,30)(27,34)(28,37)(38,39);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,2,4,3)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,3,4,2)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=f,d^2=gi,e^2=i,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 480218);