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 | |
4525 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)(3,4)(5,7)(6,8)(9,10)(11,18)(12,22)(13,16)(14,17)(15,20)(19,23)(21,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,38)(34,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(2,7)(4,6)(9,10)(11,15)(12,22)(13,16)(14,17)(18,20)(19,24)(21,23)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40);(1,4,5,6)(2,8,7,3)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(26,32)(27,35)(31,36)(34,37);(2,7)(3,8)(11,20)(15,18)(19,21)(23,24)(26,32)(30,33)(31,36)(39,40);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)Polycyclic presentation:
2^9|c^2=i,d^2=gi,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, 8012110);