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 | #37828 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4285 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)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,26,30,31)(27,28,35,33)(29,32,37,39)(34,36,38,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,26,30,31)(27,33,35,28)(29,32,37,39)(34,40,38,36);(11,13)(16,17)(19,21)(23,24)(26,31)(28,33)(29,37)(34,38);(1,2,5,4)(3,6,8,7)(9,12,15,10)(11,16,13,17)(14,18,22,20)(19,24,21,23)(25,27)(26,28)(29,38)(30,35)(31,33)(32,40)(34,37)(36,39);(1,2,5,4)(3,6,8,7)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=g,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, 7984528);