#586358

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

1710 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)(7,11)(8,9)(10,12)(13,14,17,19)(15,16,20,18)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35);(6,8)(7,12)(13,14)(15,18)(16,20)(17,19)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(5,7,9,12)(6,10,8,11)(13,15,17,20)(14,18,19,16)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,20)(18,19)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,8,9,6)(7,11,12,10)(13,16)(14,15)(17,18)(19,20)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=fg,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, 493554);