#585725

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=fg,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, 492921);