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 | |
4639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,26,31,34)(27,29,32,28)(30,33,38,40)(35,37,39,36);(1,3,5,8)(2,6,4,7)(9,11,14,15)(10,12,17,19)(13,18,22,23)(16,20,21,24)(25,27,31,32)(26,28,34,29)(30,35,38,39)(33,36,40,37);(11,15)(12,19)(18,23)(20,24)(25,28,31,29)(26,32,34,27)(30,37,38,36)(33,35,40,39);(1,4,5,2)(3,7,8,6)(9,10)(11,12)(13,21)(14,17)(15,19)(16,22)(18,24)(20,23)(25,29,31,28)(26,32,34,27)(30,36,38,37)(33,35,40,39);(9,12)(10,15)(11,17)(13,20)(14,19)(16,23)(18,21)(22,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7980334);