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 |
| prime radical | #52277 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4403 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)(9,10,14,17)(11,15,13,22)(12,19,20,16)(18,24,21,23)(25,26)(27,31)(28,32)(29,37)(30,33)(34,38)(35,39)(36,40);(2,5)(6,8)(9,11,14,13)(10,15,17,22)(12,18,20,21)(16,23,19,24)(25,27,30,29)(26,31,33,37)(28,35,36,34)(32,39,40,38);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,19)(22,24)(25,28)(26,32)(27,34)(29,35)(30,36)(31,38)(33,40)(37,39);(12,20)(16,19)(18,21)(23,24)(25,29,30,27)(26,31,33,37)(28,34,36,35)(32,39,40,38);(9,13,14,11)(10,15,17,22)(12,21,20,18)(16,23,19,24)(28,36)(32,40)(34,35)(38,39);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=i,e^2=h,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, 7610093);