Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235354 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,23)(22,24);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23);(3,8)(5,6)(9,11)(10,15)(12,21)(13,20)(14,22)(16,24)(17,19)(18,23);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,23,24)Polycyclic presentation:
2^9|d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,d^c=dhi,e^c=ei,f^c=fhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213763, #72213860, #72214564, … |
| maximal normal subgroup of | #72213763, #72213860, #72214564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142561);