#587223

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

1710 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)(7,10)(8,9)(11,12)(13,14,17,16)(15,19,20,18)(23,29)(24,31)(26,32)(27,34);(6,9)(10,12)(14,16)(18,19)(21,22,25,28)(23,26,29,32)(24,27,31,34)(30,33,35,36);(7,11)(10,12)(13,14,17,16)(15,19,20,18)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,7)(6,10)(8,11)(9,12)(13,15,17,20)(14,18,16,19)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,19,20,18)(21,24,25,31)(22,27,28,34)(23,30,29,35)(26,33,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=gh,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, 494419);