#585587

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1898 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)(5,6,9,8)(7,10,11,12)(13,14,16,18)(15,17,20,19)(22,27)(23,29)(25,33)(32,35);(6,8)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,26,27)(23,28,29,24)(25,34,33,30)(31,32,36,35);(7,11)(10,12)(13,14,16,18)(15,17,20,19)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,34)(33,36);(5,7)(6,10)(8,12)(9,11)(13,14)(15,17)(16,18)(19,20)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,8,9,6)(7,12,11,10)(13,15,16,20)(14,17,18,19)(21,25,26,33)(22,30,27,34)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=f,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, 492783);