Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,17,21,18,22,16,19)(15,23,24,20,26,27,25,28);(1,2,3,4)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(6,8)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(10,12)(13,16,18,17)(14,19,22,21)(15,24,26,25)(20,27,28,23)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,d^b=di,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264145, #72264158, #72264208, … |
| maximal normal subgroup of | #72264145, #72264158, #72264208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299836);