Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,24,22,20)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(1,3,5,8)(2,6,7,4)(9,11,14,16)(10,13,12,15)(17,19,21,22)(18,20,23,24)(26,31)(27,34)(29,37)(36,39);(2,7)(4,6)(10,12)(11,16)(17,20,21,24)(18,22,23,19)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,4,5,6)(2,3,7,8)(9,12,14,10)(11,13,16,15)(17,20)(18,19)(21,24)(22,23)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=fghi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1820361);