Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,18,13,20,10,17)(7,21,24,28,26,30,23,36)(8,27,29,15,31,22,14,25)(9,32,33,35,34,19,12,16);(5,7,12,25,10,23,34,22,13,26,33,15,11,24,9,27)(6,14,19,28,18,8,16,30,20,29,32,36,17,31,35,21);(2,4)(5,8)(6,15)(7,16)(9,28)(10,29)(11,14)(12,30)(13,31)(17,22)(18,27)(19,24)(20,25)(21,33)(23,32)(26,35)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=di,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250371, #72250385, #72250392, … |
| maximal normal subgroup of | #72250371, #72250385, #72250392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53651);