#582533

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1906 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)(5,6,9,7)(8,11,12,10)(14,17)(15,19)(21,22)(23,28)(24,30)(25,33)(26,27)(29,34)(31,32)(35,36);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,14,16,17)(15,18,19,20)(22,27)(23,30)(29,33)(31,36);(6,7)(10,11)(13,14,16,17)(15,18,19,20)(21,23,26,30)(22,28,27,24)(25,31,34,36)(29,35,33,32);(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,24,26,28)(22,23,27,30)(25,32,34,35)(29,31,33,36);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)(21,25,26,34)(22,29,27,33)(23,31,30,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=f,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, 489729);