Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,19)(12,16)(13,26)(14,18)(15,22)(17,27)(20,33)(21,34)(23,37)(24,31)(25,40)(28,35)(29,36)(30,38)(32,39);(9,11)(10,15,18,28)(12,23)(13,22)(14,20)(16,30,31,29)(17,19,26,33)(21,24)(25,38)(27,35)(32,37,40,34)(36,39);(2,4)(11,20)(12,24)(15,28)(16,31)(19,33)(22,35)(25,39)(32,40);(9,12,13,25,14,24,27,39)(10,16,26,40,18,31,17,32)(11,21,22,36,20,23,35,38)(15,29,33,37,28,30,19,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eh,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=di,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262110);