Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3648 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)(6,8)(9,10,13,16,14,17,21,12)(11,15,18,23,20,22,24,19);(2,4)(10,12)(11,18)(13,21)(16,17)(19,23)(20,24);(5,6,7,8)(9,11,13,18,14,20,21,24)(10,15,16,23,17,22,12,19);(9,12,21,17,14,16,13,10)(11,19,24,22,20,23,18,15);(11,20)(15,22)(18,24)(19,23)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=hi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226904, #72226911, #72226931, … |
| maximal normal subgroup of | #72226904, #72226911, #72226931, … |
| maximal quotient of | #72238785, #72315942 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372840);