Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24)(25,26)(27,33)(28,36)(29,32)(30,34)(31,37)(35,38)(39,40);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(27,34)(30,33)(35,40)(38,39);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(12,20)(15,19)(17,24)(21,23)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39);(10,16)(14,22)(15,19)(21,23)(26,32)(27,34)(31,36)(35,40);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|b^2=hi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610068);