#576208

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1918 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=h,e^2=f,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, 483404);