Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 | |
4717 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)(11,16)(12,17)(13,15)(14,18)(19,23)(20,21)(22,24);(2,4)(10,15)(11,17)(12,21)(14,23)(16,22)(20,24);(6,8)(9,11)(10,16)(12,19)(13,20)(14,24)(15,21)(17,23)(18,22);(10,15)(11,17)(12,21)(14,23)(16,22)(20,24);(9,12,13,22)(10,17,15,24)(11,19,20,18)(14,16,23,21)Polycyclic presentation:
2^9|e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313514, #72313623, #72315333 |
| maximal normal subgroup of | #72313514, #72313623, #72315333 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480677);