Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60951 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,14,17,15,11,13,18)(12,16,20,24,22,19,21,23);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24);(9,12,14,20,15,22,13,21)(10,16,17,24,11,19,18,23);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,17,11,18)(12,20,22,21)(16,23,19,24);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,18,11,17)(12,21,22,20)(16,23,19,24)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,d^b=di,f^b=fh,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286982, #72319828, #72320183 |
| maximal normal subgroup of | #72286982, #72319828, #72320183 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252694);