Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4350 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)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(5,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,23)(20,24);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,f^b=fi,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72211251, #72234288, #72240015, … |
| maximal normal subgroup of | #72211251, #72234288, #72240015, … |
| maximal quotient of | #72211189, #72234296, #72240021 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260633);