Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,17)(11,18,20,15)(12,16,22,13)(19,21,24,23)(25,26)(27,35)(28,33)(29,32)(30,34)(31,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,34,35)(28,37,29,36)(32,40,33,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,17)(14,22)(15,23)(18,21)(20,24)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(2,6)(3,8)(10,17)(12,22)(15,18)(19,24)(25,29,30,28)(26,33,34,32)(27,37,38,36)(31,40,35,39);(2,6)(5,7)(11,20)(15,18)(19,24)(21,23)(26,34)(31,35)(32,33)(39,40);(9,13,14,16)(10,12,17,22)(11,21,20,23)(15,19,18,24)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012334);