#568744

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

2054 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)(5,6,8,9)(7,11,12,10)(13,14,17,19)(15,16,20,18)(21,22)(23,26)(24,28)(25,33)(27,30)(29,35)(31,36)(32,34);(6,9)(7,12)(13,15)(14,18)(16,19)(17,20)(21,23,27,24)(22,28,30,26)(25,31,35,32)(29,36,33,34);(1,2,4,3)(6,9)(10,11)(14,19)(15,20)(21,24,27,23)(22,28,30,26)(25,31,35,32)(29,34,33,36);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,19)(18,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33);(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,18)(19,20)(21,26,27,28)(22,23,30,24)(25,34,35,36)(29,31,33,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=gi,e^2=fh,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, 475940);