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 | |
4406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(26,31)(28,36)(29,37)(32,38);(2,3)(6,7)(10,16)(11,12)(18,23)(19,20)(26,31)(27,35)(28,36)(34,40);(1,2,5,3)(4,6,8,7)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,26,30,31)(27,32,35,38)(28,33,36,39)(29,34,37,40);(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,27)(26,32)(28,29)(30,35)(31,38)(33,34)(36,37)(39,40);(1,3,5,2)(4,7,8,6)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,28)(26,33)(27,29)(30,36)(31,39)(32,34)(35,37)(38,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,29)(26,34)(27,28)(30,37)(31,40)(32,33)(35,36)(38,39)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017191);