Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213752 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,13,22,28)(7,15,18,29,17,11,21,24)(8,12,19,27,20,26,33,36)(16,23,31,34,30,25,32,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,21)(10,17)(12,25)(13,15)(14,29)(18,22)(19,32)(20,30)(23,26)(24,28)(27,34)(31,33)(35,36);(2,4)(5,8,10,20)(6,12,13,26)(7,16,17,30)(9,19,22,33)(11,23,15,25)(14,27,28,36)(18,31,21,32)(24,34,29,35);(6,13)(9,22)(12,26)(18,21)(19,33)(24,29)(31,32)(34,35)Polycyclic presentation:
2^9|a^2=h,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284899, #72286224, #72286231 |
| maximal normal subgroup of | #72284899, #72286224, #72286231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290339);