Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,10)(7,8)(11,12)(13,14)(15,17)(16,18)(19,20);(5,6,7,10,9,12,8,11)(13,15)(14,17)(16,20)(18,19);(6,10)(7,8)(11,12)(13,14)(15,19,20,17)(16,18)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263993, #72264008, #72264051, … |
| maximal normal subgroup of | #72263993, #72264008, #72264051, … |
| maximal quotient of | #72279865, #72281431, #72281697 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28797);