Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1444 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,13,18,14,17,25,31)(11,15,21,28,20,27,34,38)(12,16,24,29,23,30,36,39)(19,32,33,40,22,26,35,37);(1,2,3,6,5,8,4,7)(9,11)(10,15)(12,22)(13,21)(14,20)(16,26)(17,27)(18,28)(19,23)(24,35)(25,34)(29,37)(30,32)(31,38)(33,36)(39,40);(2,6)(3,4)(7,8)(9,12)(10,16)(11,19,20,22)(13,24)(14,23)(15,26,27,32)(17,30)(18,29)(21,33,34,35)(25,36)(28,37,38,40)(31,39)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224907, #72224912, #72224913, … |
| maximal normal subgroup of | #72224907, #72224912, #72224913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28800);