#575621

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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

3126 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=fhi,d^2=f,e^2=fg,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, 482817);