Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33582 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(10,14,16,23)(11,17)(12,20)(15,19,21,24)(18,22);(1,2,5,6)(3,7,4,8)(10,15,16,21)(11,18)(14,19,23,24)(17,22);(1,3,5,4)(2,7,6,8)(9,10)(11,19)(12,21)(13,16)(14,22)(15,20)(17,23)(18,24);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=e,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72264735, #72264749, #72264833, … |
| maximal normal subgroup of | #72264735, #72264749, #72264833, … |
| maximal quotient of | #72309602, #72309654, #72309852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405660);