Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #394 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,21)(12,16)(13,20)(14,18)(15,19)(17,23)(22,24);(1,2)(3,4)(9,11,14,23,15,22,13,12)(10,16,20,24,19,17,18,21);(1,3,4,2)(9,12,13,22,15,23,14,11)(10,17,18,24,19,16,20,21)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72227638, #72235489, #72240060 |
| maximal normal subgroup of | #72227638, #72235489, #72240060 |
| maximal quotient of | #72235537, #72240033, #72240034 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30925);