Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4674 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)(3,4)(5,6)(7,8)(9,10,12,16,14,17,13,21)(11,15,20,23,19,24,18,22);(2,4)(9,11,13,20,14,19,12,18)(10,15,16,22,17,24,21,23);(6,8);(9,12,14,13)(10,16,17,21)(11,18,19,20)(15,22,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72251260, #72251261, #72251265, … |
| maximal normal subgroup of | #72251260, #72251261, #72251265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260612);