#583226

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2066 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)(6,9)(7,11)(13,14)(15,19)(16,17)(18,20)(21,22,27,30)(23,28,24,25)(26,34,35,29)(31,33,32,36);(5,6)(7,10)(8,9)(11,12)(14,17)(18,19)(21,23,27,24)(22,25,30,28)(26,31,35,32)(29,33,34,36);(1,3,4,2)(6,9)(10,12)(13,14,16,17)(15,18,20,19)(21,24,27,23)(22,28,30,25)(26,32,35,31)(29,36,34,33);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,25,27,28)(22,24,30,23)(26,33,35,36)(29,32,34,31);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,18,17,19)(21,26,27,35)(22,29,30,34)(23,31,24,32)(25,33,28,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fghi,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 490422);