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 | |
4820 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)(10,14)(11,18)(15,17)(19,20);(2,4)(9,10,12,14)(11,16,18,22)(13,15,21,17)(19,23,20,24);(6,8)(10,15)(11,19)(13,21)(14,17)(18,20)(23,24);(9,11)(10,16)(12,18)(13,19)(14,22)(15,23)(17,24)(20,21);(10,15)(11,19)(13,21)(14,17)(18,20)(23,24)Polycyclic presentation:
2^9|b^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 | #72249507, #72249616, #72249632, … |
| maximal normal subgroup of | #72249507, #72249616, #72249632, … |
| maximal quotient of | #72264302, #72264323, #72264505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480674);