Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C43 |
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 | |
1086 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)(19,29)(20,31)(21,22)(24,25)(26,27)(30,32);(2,4)(5,7,8,6)(17,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,31)(29,32);(10,13)(14,16)(17,19,22,20,23,31,21,29)(18,24,27,32,28,30,26,25);(9,11,12,15)(10,14,13,16)(17,20,21,19,23,29,22,31)(18,25,26,30,28,32,27,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300768);