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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,26,30,35)(27,31,36,32)(28,33,38,39)(29,37,40,34);(2,6)(3,7)(10,14)(11,18)(16,22)(19,21)(25,27)(26,31)(28,37)(29,39)(30,36)(32,35)(33,40)(34,38);(1,3)(2,4)(5,7)(6,8)(12,20)(16,22)(19,21)(23,24)(25,27,30,36)(26,32,35,31)(28,34,38,37)(29,33,40,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,20)(22,23)(25,28)(26,33)(27,34)(29,31)(30,38)(32,40)(35,39)(36,37);(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,27)(26,31)(28,34)(29,33)(30,36)(32,35)(37,38)(39,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,29)(26,34)(27,33)(28,31)(30,40)(32,38)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=i,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231857);