Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #394 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2550 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)(17,28)(22,37)(24,33)(25,32)(27,31)(29,35)(30,34)(36,38)(39,40);(2,4)(9,11)(10,16)(12,27,29,22)(13,24)(14,23)(15,25)(17,31,34,37)(18,32)(19,21)(20,33)(26,40,35,36)(28,38,30,39);(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,26)(19,35)(20,34)(21,36)(23,38)(24,39)(25,27)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,e^c=ei,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314501, #72314504, #72314514, … |
| maximal normal subgroup of | #72314501, #72314504, #72314514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30835);