#576051

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=g,e^2=f,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, 483247);