Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12)(7,10,8,11)(14,20)(16,22)(17,18)(21,23)(24,28)(25,26);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,14,17,20,19,16,18,22)(15,23,25,21,27,28,26,24);(5,7,9,8)(6,11,12,10)(13,15)(14,21)(16,24)(17,26)(18,25)(19,27)(20,23)(22,28);(2,4)(5,8,9,7)(6,11,12,10)(13,16,19,14)(15,24,27,21)(17,20,18,22)(23,25,28,26)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339434);