Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,25)(10,15)(11,14)(12,18)(13,23)(17,32)(19,27)(20,28)(22,29)(24,30)(26,34)(31,33)(35,36);(5,7,10,18,11,20,9,19)(6,12,15,28,14,27,25,16)(8,17,23,33,24,34,22,35)(13,26,30,36,29,32,21,31);(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,26)(14,29)(15,30)(16,31)(18,33)(19,35)(20,34)(21,25)(27,32)(28,36);(1,3,4,2)(6,14)(12,27)(13,29)(15,25)(16,28)(21,30)(26,32)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,e^b=ei,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72214887, #72224693, #72312227, … |
| maximal normal subgroup of | #72214887, #72224693, #72312227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261443);