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 | |
4315 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)(4,8)(5,6)(9,10,14,13)(11,17,19,15)(12,20,22,16)(18,23,21,24)(26,31)(28,34)(32,38)(36,40);(2,6)(3,8)(9,11)(10,15)(12,21)(13,17)(14,19)(16,24)(18,22)(20,23)(27,35)(32,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(10,13)(15,17)(16,20)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,27,30,35)(26,32,31,38)(28,36,34,40)(29,37,33,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,21)(13,17)(14,19)(16,24)(18,22)(20,23)(25,28,30,34)(26,33,31,29)(27,36,35,40)(32,39,38,37);(9,13,14,10)(11,17,19,15)(12,20,22,16)(18,24,21,23)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,40,38,36)Polycyclic presentation:
2^9|a^2=gi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231738);