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 | |
4650 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)(11,16)(12,20)(13,17)(14,18)(15,22)(19,21)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,38)(36,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,19)(14,17)(15,21)(16,18)(20,23)(22,24)(26,28)(27,32)(33,36)(35,40);(1,3)(2,5)(4,8)(6,7)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,38)(39,40);(9,12,14,22)(10,15,18,20)(11,19,17,24)(13,21,16,23)(25,28,31,26)(27,34,32,29)(30,36,39,33)(35,38,40,37);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,19)(13,18)(14,17)(15,23)(20,21)(22,24)(25,29,31,34)(26,32,28,27)(30,38,39,37)(33,35,36,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,16)(10,17,18,11)(12,21,22,23)(15,24,20,19)(25,30,31,39)(26,33,28,36)(27,35,32,40)(29,37,34,38)Polycyclic presentation:
2^9|d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003074);