Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,10)(7,8)(11,12)(13,14)(15,19,20,17)(16,18);(5,6,8,12,9,11,7,10)(13,15)(14,17)(16,20)(18,19);(1,2,3,4)(6,11)(10,12)(15,20)(17,19)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=hi,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264005, #72264015, #72264051, … |
| maximal normal subgroup of | #72264005, #72264015, #72264051, … |
| maximal quotient of | #72280067, #72280891 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12506);