Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,12,8,10,15,13)(7,14,16,19,18,20,17,11);(1,2)(3,4)(6,10)(12,13);(2,4)(5,7,15,16,8,18,9,17)(6,11,13,14,10,19,12,20);(5,6)(7,11)(8,10)(9,12)(13,15)(14,16)(17,20)(18,19)Polycyclic presentation:
2^9|a^2=f,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^d=gi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72252649, #72252967, #72256917, … |
| maximal normal subgroup of | #72252649, #72252967, #72256917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332205);