#582346

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

1778 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=h,e^2=g,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, 489542);