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 | |
4676 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)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,38)(39,40);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(17,20)(19,24)(26,28)(27,32)(33,36)(35,40);(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,28,31,26)(27,34,32,29)(30,36,39,33)(35,38,40,37);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,29,31,34)(26,32,28,27)(30,38,39,37)(33,35,36,40);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,30,31,39)(26,33,28,36)(27,35,32,40)(29,37,34,38)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7809253);