#582254

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^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, 489450);