Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(18,22)(19,25)(20,26)(23,28)(24,30)(27,31);(1,2,3,4)(6,8)(10,13)(14,15)(17,18,20,22,21,23,26,28)(19,24,25,29,27,30,31,32);(9,11,12,16)(10,14,13,15)(18,23)(22,28)(24,30)(29,32);(17,19,26,31,21,27,20,25)(18,24,28,32,23,30,22,29)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,d^2=gi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301059);