Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#433 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,11,20,18)(12,16,25,32,24,21,34,31)(13,17,28,33,29,22,27,26)(23,30,39,40,37,35,38,36);(9,11)(10,15)(12,21)(13,26)(14,18)(16,24)(17,28)(19,20)(22,27)(23,36)(25,31)(29,33)(30,39)(32,34)(35,38)(37,40);(2,5)(6,8)(9,12)(10,16)(11,21)(13,23)(14,25)(15,24)(17,30)(18,31)(19,32)(20,34)(22,35)(26,36)(27,38)(28,39)(29,37)(33,40);(9,13)(10,17)(11,22)(12,23)(14,28)(15,29)(16,30)(18,26)(19,33)(20,27)(21,35)(24,37)(25,39)(31,36)(32,40)(34,38);(1,3,4,7)(2,6,5,8)(12,24)(16,21)(23,37)(25,34)(30,35)(31,32)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72313439, #72313447, #72313507, … |
| maximal normal subgroup of | #72313439, #72313447, #72313507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288190);