#583109

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

1726 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)(15,19)(18,20)(21,22,26,24)(23,28,31,27)(25,32,33,29)(30,34,36,35);(5,6,8,9)(7,10,11,12)(14,17)(18,20)(21,23)(22,27)(24,28)(25,30)(26,31)(29,34)(32,35)(33,36);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,34)(33,36);(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,18,17,20)(21,24,26,22)(23,28,31,27)(25,32,33,29)(30,35,36,34);(5,7)(6,10)(8,11)(9,12)(21,25,26,33)(22,29,24,32)(23,30,31,36)(27,34,28,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=fgh,e^2=fg,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, 490305);