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 | |
4828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,11)(15,17)(16,18)(22,23)(25,26)(27,36)(28,39)(29,40)(30,35)(31,33)(32,37)(34,38);(1,2,5,6)(3,7,4,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,27)(26,32)(28,35)(29,38)(30,34)(31,37)(33,36)(39,40);(10,11)(12,19)(16,18)(20,24)(26,33)(27,37)(29,35)(34,39);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,31,38)(26,34,33,39)(27,35,37,29)(30,36,40,32);(1,4,5,3)(2,8,6,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,35)(26,30,33,40)(27,38,37,28)(32,34,36,39);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,35)(27,39)(28,36)(29,33)(31,40)(32,38)(34,37)Polycyclic presentation:
2^9|b^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646714);