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 | |
4556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2)(3,6)(4,7)(5,8)(10,15)(12,18)(17,21)(20,24)(27,36)(29,37)(32,35)(34,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,16)(13,22)(20,24)(25,27,31,36)(26,32,30,35)(28,37,39,29)(33,34,38,40);(3,8)(5,6)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,28,31,39)(26,33,30,38)(27,29,36,37)(32,40,35,34);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,35)(36,39);(1,2)(3,6)(4,7)(5,8)(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|c^2=gh,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8811700);