Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2311 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13958 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1123 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)(4,17,20,26)(5,10)(6,11)(7,12)(8,15)(9,19,27,18)(14,22,30,25)(16,23)(21,24)(28,29,31,32);(1,3)(2,8,12,24)(4,14)(5,15)(6,21)(7,16)(9,25,27,22)(10,23,11,13)(17,31,26,28)(18,29)(19,32)(20,30);(1,4,5,18,7,20,6,19)(2,9,11,17,12,27,10,26)(3,14,15,29,16,30,21,32)(8,22,13,28,24,25,23,31);(2,10,12,11)(8,23,24,13)(9,26,27,17)(22,31,25,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72306445, #72306448, #72306473, … |
| maximal normal subgroup of | #72306445, #72306448, #72306473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271256);