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 |
| prime radical | #38026 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,12)(11,17,13,16)(14,20,22,18)(19,23,21,24)(25,26,31,28)(27,35,36,32)(29,33,38,39)(30,40,37,34);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,17)(13,15)(14,19)(18,23)(20,24)(21,22)(25,27,31,36)(26,32,28,35)(29,30,38,37)(33,34,39,40);(10,12)(11,13)(18,20)(19,21)(27,36)(29,38)(32,35)(33,39);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,26)(27,35,36,32)(29,39,38,33)(30,40,37,34);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,31,38)(26,33,28,39)(27,37,36,30)(32,40,35,34);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30,31,37)(26,34,28,40)(27,38,36,29)(32,39,35,33)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7627528);