Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,16,11,14,9,15)(7,17,20,28,21,27,19,12)(8,22,24,30,25,31,23,13)(18,29,34,36,35,26,33,32);(5,7,9,20,11,21,10,19)(6,12,16,27,14,28,15,17)(8,18,24,33,25,35,23,34)(13,29,31,36,30,26,22,32);(1,3,4,2)(5,8)(6,13)(7,18,21,35)(9,23)(10,24)(11,25)(12,26,28,29)(14,30)(15,31)(16,22)(17,32,27,36)(19,33,20,34);(6,14)(7,19)(9,10)(12,17)(13,30)(18,33)(20,21)(23,24)(26,32)(27,28)(29,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,h^a=hi,c^b=cef,d^b=de,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286350, #72286356, #72286383, … |
| maximal normal subgroup of | #72286350, #72286356, #72286383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264568);