#585170

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

1862 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)(5,6,9,12)(7,8,11,10)(15,19)(17,20)(21,22)(23,29)(24,32)(25,26)(27,30)(28,33)(31,34)(35,36);(1,2,4,3)(7,11)(8,10)(13,14,16,18)(15,17,19,20)(22,26)(23,30)(28,32)(31,36);(5,7,9,11)(6,10,12,8)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,35)(33,36);(5,8,9,10)(6,11,12,7)(13,15,16,19)(14,17,18,20)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(1,3,4,2)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=fh,e^2=fgi,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 492366);