Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,13)(7,10,14,18)(8,11,16,12)(15,17,20,19)(21,22,25,29,26,24,30,28)(23,27,32,35,33,31,36,34);(1,2,4,3)(5,7)(6,10)(8,15)(9,14)(11,17)(12,19)(13,18)(16,20)(21,23,30,36,26,33,25,32)(22,27,28,34,24,31,29,35);(1,3,4,2)(6,11)(8,16)(10,17)(12,13)(15,20)(18,19)(21,24)(22,26)(23,31)(25,28)(27,33)(29,30)(32,34)(35,36);(7,14)(10,18)(15,20)(17,19)(22,24)(27,31)(28,29)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dhi,d^c=di,f^c=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109665);