Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,20)(12,17)(13,27)(14,30)(15,19)(16,22)(18,28)(21,35)(23,33)(24,39)(25,32)(26,34)(29,40)(31,37)(36,38);(9,11,13,23,12,21,26,38,15,25,29,22,14,24,28,37)(10,16,18,32,30,36,40,35,19,33,34,20,17,31,27,39);(2,5)(6,8)(9,12,15,14)(10,17,19,30)(11,21,25,24)(13,26,29,28)(16,31,33,36)(18,27,34,40)(20,35,32,39)(22,37,23,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72309634, #72309644, #72309649, … |
| maximal normal subgroup of | #72309634, #72309644, #72309649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33894);