Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1240 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)(15,24)(16,18)(19,27)(20,28)(21,23)(25,26);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,21)(17,22)(18,23)(19,25)(24,28)(26,27);(2,4)(5,8,9,6)(7,11,12,10)(13,15)(14,19)(16,26)(17,25)(18,24)(20,22)(21,28)(23,27);(7,12)(10,11)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,28)(23,27)Polycyclic presentation:
2^9|a^2=e,b^2=gi,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345086);