Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | |
1482 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)(9,10)(11,15)(12,13)(14,16)(17,18,22,25,23,26,21,19)(20,28,29,24,31,27,30,32);(5,6,7,8)(10,13)(14,15)(17,19,21,26,23,25,22,18)(20,24,30,28,31,32,29,27);(2,4)(6,8)(17,20,23,31)(18,24,26,32)(19,27,25,28)(21,29,22,30);(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,19,26,25)(20,29,31,30)(24,27,32,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269097);