Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23);(2,4)(9,11,14,20)(10,15,18,24)(12,19,13,21)(16,23,17,22);(10,16)(12,13)(15,22)(17,18)(19,21)(23,24);(2,4)(9,12,14,13)(10,17,18,16)(11,19,20,21)(15,22,24,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250619, #72250648, #72250788, … |
| maximal normal subgroup of | #72250619, #72250648, #72250788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266664);