Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,24,26,30,25,29)(23,31,35,36,34,28,33,32);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,26,34)(22,28,30,31)(24,32,29,36)(25,33,27,35);(2,4)(21,24,27,30,26,29,25,22)(23,32,35,31,34,36,33,28);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,25,26,27)(22,29,30,24)(23,33,34,35)(28,36,31,32);(6,12)(10,13)(11,20)(18,19)(22,30)(23,34)(24,29)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6299948);