Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34633 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(15,21,22,23)(18,25,26,28);(1,2,3,4)(6,9)(10,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,26)(23,28)(24,27);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,23)(17,22)(19,26)(20,28)(21,24)(25,27);(7,11)(10,12)(14,19)(18,26)(20,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,f^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287713);