#590621

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

1686 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)(5,6,9,11)(7,8,12,10)(15,19)(18,20)(21,22)(23,27)(24,31)(25,26)(28,33)(29,34)(30,36)(32,35);(5,7,9,12)(6,10,11,8)(14,17)(18,20)(22,26)(23,29)(28,31)(30,32);(1,3,4,2)(6,11)(7,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,29)(22,27,26,34)(24,32,33,30)(28,36,31,35);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,24)(22,28)(23,30)(25,33)(26,31)(27,35)(29,32)(34,36);(5,8,9,10)(6,7,11,12)(21,22)(23,27)(24,28)(25,26)(29,34)(30,35)(31,33)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=fh,e^2=g,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, 497817);