Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C4×#95 |
| prime radical | C24 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10,12,15,14,18,13,17)(11,16,20,22,19,24,21,23);(2,6)(3,4)(10,15,18,17)(11,19)(16,23,24,22)(20,21);(1,3,5,4)(2,7,6,8)(9,11,13,21,14,19,12,20)(10,16,17,23,18,24,15,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,e^2=h,g^2=hi,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72265314, #72265468, #72265478, … |
| maximal normal subgroup of | #72265314, #72265468, #72265478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12524);