Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395166 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(10,16)(11,19)(12,18)(13,14)(17,20)(22,23);(1,2,3,4)(5,6)(7,8)(9,10,14,12,15,18,13,16)(11,20,23,24,22,17,19,21);(6,8)(9,11)(10,17)(12,21)(13,19)(14,23)(15,22)(16,24)(18,20);(9,12,13,10,15,16,14,18)(11,21,19,17,22,24,23,20)Polycyclic presentation:
2^9|b^2=ei,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226904, #72226922, #72226931, … |
| maximal normal subgroup of | #72226904, #72226922, #72226931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319029);