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 | |
12782 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)(9,10,15,19)(11,16,20,29)(12,13,17,26)(14,27,28,18)(21,22,30,33)(23,37,34,31)(24,38,32,25)(35,40,39,36);(5,6,8,7)(11,20)(16,29)(21,30)(22,33)(23,34)(31,37)(35,39)(36,40);(1,2,3,4)(5,7,8,6)(9,11)(10,16)(12,21)(13,22)(14,23)(15,20)(17,30)(18,31)(19,29)(24,35)(25,36)(26,33)(27,37)(28,34)(32,39)(38,40);(9,12)(10,13)(11,21)(14,24)(15,17)(16,22)(18,25)(19,26)(20,30)(23,35)(27,38)(28,32)(29,33)(31,36)(34,39)(37,40);(9,13)(10,17)(11,22)(12,19)(14,25)(15,26)(16,30)(18,32)(20,33)(21,29)(23,36)(24,27)(28,38)(31,39)(34,40)(35,37);(9,14,15,28)(10,18,19,27)(11,23,20,34)(12,24,17,32)(13,25,26,38)(16,31,29,37)(21,35,30,39)(22,36,33,40)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^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538128);