Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3295 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)(11,20)(12,22)(13,18)(14,24)(15,19)(16,17)(21,23);(5,6)(7,8)(9,11,13,21,15,22,14,17)(10,16,24,12,19,23,18,20);(2,4)(6,8)(9,12,14,16,15,20,13,23)(10,17,18,22,19,21,24,11);(6,8)(9,13,15,14)(10,18,19,24)(11,17,22,21)(12,23,20,16);(9,13,15,14)(10,18,19,24)(11,21,22,17)(12,16,20,23)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315800, #72315838, #72316003 |
| maximal normal subgroup of | #72315800, #72315838, #72316003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346162);