Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,15,19,16,20,14,18)(11,21,17,24,22,12,23,13);(5,6)(7,8)(9,11)(10,13)(12,19)(14,17)(15,23)(16,22)(18,21)(20,24);(2,4)(6,8)(9,12,16,21)(10,17,20,23)(11,19,22,18)(13,14,24,15);(6,8)(9,13)(10,11)(12,14)(15,21)(16,24)(17,19)(18,23)(20,22);(9,14,16,15)(10,18,20,19)(11,17,22,23)(12,13,21,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315800, #72315838, #72315877, … |
| maximal normal subgroup of | #72315800, #72315838, #72315877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344974);