Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,11,8,10)(15,19)(17,20)(21,22,26,23)(24,29,32,27)(25,28,34,30)(31,36,33,35);(13,14,16,18)(15,17,19,20)(22,23)(27,29)(28,30)(35,36);(5,7,9,8)(6,10,12,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,22)(24,29,32,27)(25,30,34,28)(31,36,33,35);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,24)(22,27)(23,29)(25,33)(26,32)(28,36)(30,35)(31,34);(5,8,9,7)(6,11,12,10)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fg,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1203212);