#564539

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1830 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)(6,8)(10,12)(13,14,19,17)(15,20,18,16)(21,22,26,25)(23,27,31,30)(24,32,34,28)(29,36,33,35);(7,11)(10,12)(13,15,19,18)(14,20,17,16)(22,25)(27,30)(28,32)(35,36);(5,6,9,8)(7,10,11,12)(13,16)(14,18)(15,17)(19,20)(21,23,26,31)(22,27,25,30)(24,33,34,29)(28,36,32,35);(1,3,4,2)(5,7,9,11)(6,10,8,12)(13,17,19,14)(15,20,18,16)(21,24)(22,28)(23,29)(25,32)(26,34)(27,35)(30,36)(31,33);(5,8,9,6)(7,12,11,10)(13,18,19,15)(14,20,17,16)(21,25,26,22)(23,30,31,27)(24,32,34,28)(29,36,33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgh,d^2=fghi,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 471735);