Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #380 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10)(11,19,22,15)(12,23)(13,26)(14,17)(16,24)(18,27)(20,33,35,37)(21,29,36,32)(25,38)(28,34,31,40)(30,39);(1,2)(3,4)(5,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,19)(18,29)(23,37)(24,34)(25,35)(26,32)(27,36)(30,31)(33,38)(39,40);(2,4)(9,12,13,24,14,25,27,39)(10,16,18,23,17,30,26,38)(11,20,21,34,22,35,36,40)(15,28,29,37,19,31,32,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314509, #72314517, #72314530, … |
| maximal normal subgroup of | #72314509, #72314517, #72314530, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31075);