Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,7)(13,14,17,19,18,22,16,21)(15,20,25,26,23,27,24,28);(9,10,11,12)(14,19)(16,17)(20,26)(21,22)(24,25)(27,28);(1,2,3,4)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,28)(22,27);(5,7,8,6)(15,23)(20,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,e^b=ei,g^b=gi,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261874);