Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500676 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33517 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24);(1,3,4,8)(2,5,6,7)(9,11,13,20,14,21,22,12)(10,15,17,23,19,24,18,16);(10,16)(11,12)(13,22)(15,18)(17,24)(19,23)(20,21);(2,6)(5,7)Polycyclic presentation:
2^9|b^2=eg,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238786, #72239808, #72239843, … |
| maximal normal subgroup of | #72238786, #72239808, #72239843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413419);