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 | #59090 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(1,3,6,5)(2,7,4,8)(10,14)(15,17)(16,21)(22,23)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,3,6,5)(2,8,4,7)(12,20)(16,21)(18,24)(22,23)(27,33)(31,34)(35,37)(39,40);(1,4,6,2)(3,7,5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(1,5,6,3)(2,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=hi,e^2=h,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, 8118897);