Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| 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×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1486 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)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,29)(21,25)(22,27)(23,26)(24,31)(30,32);(10,13)(14,16)(17,19,21,29)(18,20,25,28)(22,30,23,24)(26,32,27,31);(2,4)(9,11)(10,14)(12,15)(13,16)(17,20,21,28)(18,24,25,30)(19,27,29,26)(22,32,23,31);(5,7,8,6)(11,15)(14,16)(18,25)(20,28)(26,27)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288633);