#589539

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 496735);