Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#388 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1926 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)(3,4)(5,6,10,14,11,15,9,25)(7,16,20,27,18,31,19,12)(8,21,23,13,24,28,22,29)(17,26,34,30,32,36,33,35);(2,4)(5,7)(6,12,15,27)(8,17)(9,19)(10,20)(11,18)(13,26,29,36)(14,16,25,31)(21,35,28,30)(22,33)(23,34)(24,32);(5,8,10,23,11,24,9,22)(6,13,14,28,15,29,25,21)(7,17,20,34,18,32,19,33)(12,26,16,30,27,36,31,35);(7,18)(12,27)(16,31)(17,32)(19,20)(26,36)(30,35)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72252198, #72252206, #72252208, … |
| maximal normal subgroup of | #72252198, #72252206, #72252208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261118);