Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | |
1108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,23,22,26,20,25)(19,24,29,30,28,31,27,32);(1,2)(3,4)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,21)(25,26)(28,29)(30,32);(2,4)(10,14)(13,16)(17,19,20,27,22,28,21,29)(18,24,25,32,26,31,23,30);(5,7,8,6)(19,28)(24,31)(27,29)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=eh,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=hi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266538);