Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,26,29,32)(21,22,30,25)(24,31,27,28);(1,2,3,4)(6,8)(17,19,25,31,23,29,22,28)(18,24,30,32,20,27,21,26);(10,13)(14,16)(17,20,22,30,23,18,25,21)(19,27,28,32,29,24,31,26);(9,11,12,15)(10,14,13,16)(17,21,22,20,23,30,25,18)(19,26,28,27,29,32,31,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=g,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264152, #72264192, #72264211, … |
| maximal normal subgroup of | #72264152, #72264192, #72264211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300830);