Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417778 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,8)(7,11,12,10)(13,14,20,27,18,17,19,21)(15,22,25,26,23,16,24,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,24,18,23,20,25)(14,16,21,26,17,22,27,28);(2,4)(5,8,9,6)(7,11,12,10)(13,16)(14,15)(17,23)(18,22)(19,28)(20,26)(21,25)(24,27);(7,12)(10,11)(13,17)(14,18)(15,22)(16,23)(19,27)(20,21)(24,28)(25,26)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329092);