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 | #43021 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4417 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)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(26,30)(27,33)(28,36)(38,40);(2,6)(3,7)(9,11,14,16)(10,15,18,12)(13,19,21,23)(17,22,24,20)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,3)(2,4)(5,7)(6,8)(13,21)(17,24)(19,23)(20,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(9,12)(10,16)(11,18)(13,20)(14,15)(17,23)(19,24)(21,22)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314042);