Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
10714 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)(3,4)(6,8)(9,10,15,13)(11,16,19,25)(12,17,22,23)(14,26,27,18)(20,28,31,36)(21,34,32,29)(24,37,35,30)(33,40,39,38);(5,6,7,8)(11,19)(12,22)(16,25)(17,23)(21,32)(24,35)(29,34)(30,37);(2,4)(9,11)(10,16)(12,20)(13,25)(14,21)(15,19)(17,28)(18,29)(22,31)(23,36)(24,33)(26,34)(27,32)(30,38)(35,39)(37,40);(1,2)(3,4)(9,12)(10,17)(11,20)(13,23)(14,24)(15,22)(16,28)(18,30)(19,31)(21,33)(25,36)(26,37)(27,35)(29,38)(32,39)(34,40);(9,13,15,10)(11,16,19,25)(12,23,22,17)(14,26,27,18)(20,28,31,36)(21,29,32,34)(24,37,35,30)(33,38,39,40);(9,14,15,27)(10,18,13,26)(11,21,19,32)(12,24,22,35)(16,29,25,34)(17,30,23,37)(20,33,31,39)(28,38,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581824);