Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #385 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1878 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)(9,10,13,17,15,18,14,12)(11,19,23,24,22,20,21,16);(5,6,7,8)(9,11)(10,16)(12,20)(13,23)(14,21)(15,22)(17,19)(18,24);(9,12,14,18,15,17,13,10)(11,20,21,24,22,19,23,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=di,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72235145, #72235146, #72235149, … |
| maximal normal subgroup of | #72235145, #72235146, #72235149, … |
| maximal quotient of | C2×#123492, #72235177, #72235278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30688);