Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #94871 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,15,16,18)(12,20)(17,24,23,22)(19,21);(2,4)(9,10,12,17,13,18,21,24,14,16,20,23,11,15,19,22);(6,8)(10,15,16,18)(12,20)(17,24,23,22)(19,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,f^b=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239244, #72239247, #72239256, … |
| maximal normal subgroup of | #72239244, #72239247, #72239256, … |
| maximal quotient of | #72239316, #72279532, C2×#147125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54321);