Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1917 |
| prime radical | C24 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,15,10,16,24,31,11,13,23,28,8,14,22,30)(7,17,21,32,20,29,34,36,19,12,25,27,18,26,33,35);(1,2,4,3)(5,7)(6,12,13,17)(8,18)(9,21)(10,20)(11,19)(14,29,16,26)(15,27,28,32)(22,33)(23,25)(24,34)(30,36,31,35);(1,3,4,2)(6,13)(7,18,19,20)(9,23)(12,26,17,29)(14,16)(21,34,25,33)(22,24)(27,36,32,35)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=di,f^b=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56127);