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 | |
4586 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)(9,10)(11,18)(12,14)(13,16)(15,17)(19,24)(20,23)(21,22)(29,37)(33,38)(34,39)(36,40);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,20)(13,15)(14,21)(17,18)(19,22)(23,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(1,3,5,4)(2,7,6,8)(10,17)(11,13)(12,19)(21,23)(26,31)(28,35)(33,38)(36,40);(1,3,5,4)(2,7,6,8)(9,12,15,19)(10,14,17,24)(11,20,13,22)(16,21,18,23)(25,27,30,32)(26,28,31,35)(29,34,37,39)(33,36,38,40);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,18,17,16)(12,22,19,20)(14,23,24,21)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(9,14)(10,19)(11,21)(12,17)(13,23)(15,24)(16,22)(18,20)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=ghi,e^2=g,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, 7624530);