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 | |
4685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,18)(11,16,19,15)(12,22,17,13)(20,21,23,24)(28,36)(31,38)(34,40)(37,39);(2,6)(3,7)(11,19)(15,16)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3)(2,4)(5,7)(6,8)(9,11,14,19)(10,15,18,16)(12,20,17,23)(13,21,22,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,23)(13,21)(14,19)(15,18)(17,20)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,12,14,17)(10,13,18,22)(11,20,19,23)(15,21,16,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,13)(10,17)(11,21)(12,18)(14,22)(15,23)(16,20)(19,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064270);