Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #393 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,17,15,11,13,18)(12,20,23,19,22,24,21,16);(2,4)(9,11,15,10)(12,19)(13,17,14,18)(16,22)(20,21)(23,24);(6,8)(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,20)(18,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,e^c=ei,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235149, #72235491, #72238861, … |
| maximal normal subgroup of | #72235149, #72235491, #72238861, … |
| maximal quotient of | #72235161, #72238865, #72238868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30837);