Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #390 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,27)(22,30)(24,32)(25,31)(28,34)(29,33)(35,37)(36,39)(38,40);(9,11,13,24,15,25,14,23)(10,16,19,21,20,32,18,31)(12,22,27,37,29,38,28,36)(17,30,33,35,34,40,26,39);(2,4)(9,12,14,28,15,29,13,27)(10,17,19,33,20,34,18,26)(11,22,23,36,25,38,24,37)(16,30,21,35,32,40,31,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72314512, #72314517, #72314528, … |
| maximal normal subgroup of | #72314512, #72314517, #72314528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30667);