Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(6,9)(7,11)(10,12)(13,14)(15,23,26,27)(16,19)(17,21)(18,22)(20,25,28,24);(1,2)(3,4)(5,6,8,10)(7,12,11,9)(14,19,22,21)(20,27,28,23);(2,4)(5,6)(7,12)(8,10)(9,11)(13,15,18,26)(14,20)(16,24,17,25)(19,27)(21,23)(22,28)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=deh,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=ei,d^c=di,e^c=ei,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29020);