Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2041 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
718 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)(5,6,9,15,10,17,27,16)(7,18,21,34,22,30,35,28)(8,23,25,13,26,11,24,14)(12,29,31,19,33,36,32,20);(6,11)(7,19)(8,24)(9,27)(13,15)(14,16)(17,23)(18,34)(20,22)(21,29)(25,26)(28,30)(31,32)(35,36);(2,4)(5,7,8,20,9,21,25,29,10,22,26,19,27,35,24,36)(6,12,13,30,16,32,23,34,17,33,14,18,15,31,11,28)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249951, #72249953, #72249979, … |
| maximal normal subgroup of | #72249951, #72249953, #72249979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55762);