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 | |
4591 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)(13,21)(16,23)(19,20)(22,24)(25,26)(27,34)(28,36)(29,31)(30,33)(32,38)(35,39)(37,40);(3,7)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,27,30,29)(26,31,33,34)(28,35,38,37)(32,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,17,18)(13,19,21,20)(16,22,23,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,15,17,18)(13,20,21,19)(16,22,23,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,29,30,27)(26,31,33,34)(28,37,38,35)(32,39,36,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026073);