#589310

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

1814 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,25)(23,27,29,31)(24,32,34,28)(30,36,33,35);(1,3,4,2)(6,9)(7,12)(14,16)(15,20)(22,25)(23,29)(28,32)(30,33);(6,9)(10,11)(13,14,17,16)(15,19,20,18)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19)(21,24)(22,28)(23,30)(25,32)(26,34)(27,35)(29,33)(31,36);(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,19,20,18)(21,25,26,22)(23,31,29,27)(24,32,34,28)(30,36,33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fgh,d^2=fgh,e^2=h,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, 496506);