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 | |
4589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26,30,34)(27,29,33,31)(28,32,38,40)(35,37,36,39);(2,5)(6,7)(10,15)(12,22)(16,19)(18,24)(25,27,30,33)(26,29,34,31)(28,36,38,35)(32,39,40,37);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,3,4,8)(2,6,5,7)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,38)(39,40);(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,23)(20,24)(25,28,30,38)(26,32,34,40)(27,35,33,36)(29,37,31,39);(9,13)(10,18)(11,17)(12,16)(14,23)(15,24)(19,22)(20,21)(25,29)(26,33)(27,34)(28,37)(30,31)(32,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8379600);