Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(9,10,11,12)(13,14,18,21,19,22,17,16)(15,23,25,20,27,28,26,24);(1,2,4,3)(5,6)(7,8)(13,15)(14,20)(16,28)(17,25)(18,26)(19,27)(21,23)(22,24);(1,3,4,2)(6,8)(13,16,17,22,19,21,18,14)(15,24,26,28,27,20,25,23);(13,17,19,18)(14,16,22,21)(15,25,27,26)(20,28,24,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,g^b=gi,h^b=hi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265510);