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 | |
4464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,19)(11,18,21,15)(12,17,13,16)(20,23,22,24)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,40)(36,38);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,35)(26,28,32,33)(29,38,39,34)(31,40,37,36);(2,5)(6,7)(9,12,14,13)(10,16,19,17)(11,20,21,22)(15,23,18,24)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,40,38,36);(2,5)(6,7)(9,13,14,12)(10,17,19,16)(11,20,21,22)(15,23,18,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,18)(12,22)(13,20)(14,21)(15,19)(16,23)(17,24)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,18)(12,16)(13,17)(14,19)(15,21)(20,24)(22,23)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,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, 10228026);