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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,13)(14,15)(17,21)(18,24)(19,22)(20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(2,6)(3,8)(10,15)(12,21)(16,23)(18,22)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3)(2,4)(5,8)(6,7)(10,15)(13,17)(16,23)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,22)(13,19)(14,20)(15,23)(17,24)(18,21)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,18)(13,15)(14,21)(16,19)(20,22)(23,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39);(9,13,14,17)(10,12,15,21)(11,19,20,24)(16,22,23,18)Polycyclic presentation:
2^9|b^2=gh,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325358);