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 | #49839 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,26)(27,31)(28,33)(29,39)(30,32)(34,36)(35,40)(37,38);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,27,30,35)(26,31,32,40)(28,36,38,29)(33,34,37,39);(1,2,5,4)(3,6,8,7)(10,16)(15,17)(18,23)(22,24)(26,32)(27,35)(29,36)(33,37);(1,2,5,4)(3,6,8,7)(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,28)(26,33)(27,29)(30,38)(31,39)(32,37)(34,40)(35,36);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29)(26,34)(27,28)(30,36)(31,37)(32,39)(33,40)(35,38);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,26)(27,31)(28,37)(29,34)(30,32)(33,38)(35,40)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7995469);