Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| 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×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1498 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)(19,27)(20,23)(21,24)(22,26)(25,30)(28,32)(29,31);(5,7,8,6)(10,13)(14,16)(18,23)(19,28)(24,26)(29,30);(2,4)(9,11,12,15)(10,14,13,16)(18,24)(21,22)(23,26)(25,32)(27,31)(29,30);(17,19,20,28)(18,25,23,31)(21,30,22,29)(24,32,26,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,d^b=di,f^c=fi,g^c=gi,f^d=fi,g^d=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299969);