Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13896 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,4,8)(5,19,21,10)(6,12)(7,11)(9,16,24,15)(14,23)(17,22)(18,27)(20,32,30,26)(25,29)(28,31);(1,3,6,15,7,4,17,16)(2,8,12,24,11,13,22,9)(5,14,20,28,21,18,30,29)(10,23,26,31,19,27,32,25);(1,4)(2,9,11,24)(3,7)(5,18)(6,16)(8,22,13,12)(10,25,19,31)(14,21)(15,17)(20,29)(23,32,27,26)(28,30);(1,5,6,20,7,21,17,30)(2,10,12,26,11,19,22,32)(3,14,15,28,4,18,16,29)(8,23,24,31,13,27,9,25)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319976, #72319982, #72320069, … |
| maximal normal subgroup of | #72319976, #72319982, #72320069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268580);