Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213×C4 |
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 | |
12822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,20)(11,16,21,31)(12,17,26,23)(13,18,27,22)(14,29,30,19)(24,38,35,32)(25,37,39,33)(28,36,40,34);(5,6,8,7)(11,21)(13,27)(16,31)(18,22)(24,35)(28,40)(32,38)(34,36);(1,2,3,4)(5,7,8,6)(9,11)(10,16)(12,18)(13,23)(14,24)(15,21)(17,27)(19,32)(20,31)(22,26)(25,34)(28,37)(29,38)(30,35)(33,40)(36,39);(9,12)(10,17)(11,22)(13,16)(14,25)(15,26)(18,21)(19,33)(20,23)(24,36)(27,31)(28,32)(29,37)(30,39)(34,35)(38,40);(9,13)(10,18)(11,23)(12,16)(14,28)(15,27)(17,21)(19,34)(20,22)(24,37)(25,32)(26,31)(29,36)(30,40)(33,35)(38,39);(9,14,15,30)(10,19,20,29)(11,24,21,35)(12,25,26,39)(13,28,27,40)(16,32,31,38)(17,33,23,37)(18,34,22,36)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537858);