Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,20)(16,25)(17,22)(18,28)(19,23)(21,26)(24,27);(2,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,23)(17,27)(18,21)(19,25)(22,24)(26,28);(2,4)(5,8,9,7)(6,10,11,12)(13,14)(15,24)(16,21)(17,28)(18,22)(19,23)(20,27)(25,26);(13,16,18,26,19,20,17,24)(14,21,22,25,23,27,28,15)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320634);