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 | |
4604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,4,5,2)(3,7,8,6)(10,12)(16,17)(18,20)(23,24)(26,31)(28,32)(34,37)(36,39);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7640278);