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 | |
4673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
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)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(2,7)(4,6)(9,10)(11,15)(12,22)(13,16)(14,17)(18,20)(19,24)(21,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,4,5,6)(2,8,7,3)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(28,36)(31,40)(33,35)(37,39);(2,7)(3,8)(11,20)(15,18)(19,21)(23,24)(27,34)(30,38)(33,35)(37,39);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=g,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, 8009172);