#586148

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

1754 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=h,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 493344);