Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36157 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(3,7)(5,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(1,2)(3,5)(4,6)(7,8)(10,14)(15,17)(16,20)(22,23)(25,27,30,28)(26,32,34,31)(29,35,37,36)(33,39,40,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8108799);