Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4700 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)(5,6)(7,8)(9,10,15,12)(11,16,21,13)(14,17,24,22)(18,19,23,20);(2,4)(9,11)(10,16)(12,13)(14,19)(15,21)(17,23)(18,22)(20,24);(6,8)(9,12,15,10)(11,18,21,23)(13,20,16,19)(14,17,24,22);(9,13)(10,11)(12,21)(14,18)(15,16)(17,19)(20,22)(23,24);(11,19)(13,18)(14,24)(16,23)(17,22)(20,21)Polycyclic presentation:
2^9|a^2=i,c^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313514, #72315383, #72315401 |
| maximal normal subgroup of | #72313514, #72315383, #72315401 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480689);