Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,25,23,27,21,26)(19,24,20,30,29,32,28,31);(2,4)(5,6)(7,8)(17,19,22,20,23,29,21,28)(18,24,25,30,27,32,26,31);(6,8)(10,13)(14,16)(17,20,23,28)(18,24,27,32)(19,21,29,22)(25,31,26,30);(2,4)(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,27,26)(19,20,29,28)(24,31,32,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=fgi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279480);