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 | |
4795 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)(11,17)(12,16)(13,18)(14,15)(19,23)(20,22)(21,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,4)(5,6)(7,8)(10,15)(12,20)(13,17)(19,24)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(2,6)(3,7)(10,15)(11,18)(12,20)(21,23)(25,27,30,28)(26,32,34,31)(29,35,37,36)(33,39,40,38);(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,24)(22,23)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,3,5,7)(2,4,6,8)(9,12,14,20)(10,16,15,22)(11,19,18,24)(13,21,17,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9178984);