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 | |
4639 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)(9,10)(11,17)(12,13)(14,16)(15,19)(18,22)(20,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,23)(14,19)(16,17)(18,21)(22,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,22)(21,23)(26,31)(28,32)(34,37)(36,39);(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,23)(22,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(9,12)(10,18)(11,20)(13,16)(14,22)(15,21)(17,23)(19,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,13,14,18)(10,12,16,22)(11,21,19,23)(15,24,17,20)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|c^2=gh,d^2=hi,e^2=hi,f^2=h,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, 7636390);