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 | |
4716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(13,21)(16,20)(18,24)(22,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,39)(37,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27,30,34)(26,31,32,29)(28,35,36,40)(33,38,39,37);(1,3,4,7)(2,6,5,8)(10,12)(15,17)(16,20)(22,23)(26,32)(27,34)(28,36)(37,38);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27)(26,29)(28,35)(30,34)(31,32)(33,37)(36,40)(38,39);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,27)(28,37)(30,31)(32,34)(33,35)(36,38)(39,40)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=hi,e^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626576);