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 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,27,30,24)(20,31,32,25)(21,34,35,26)(28,36,41,39)(29,37,43,40)(33,38,45,44)(42,47,48,46);(3,7)(6,8)(10,13)(14,15)(18,23)(20,32)(24,27)(26,34)(28,41)(33,45)(37,40)(42,48);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,23,27)(20,28,32,41)(21,29,35,43)(25,36,31,39)(26,37,34,40)(33,42,45,48)(38,46,44,47);(1,3,5,7)(2,6,4,8)(9,11,12,16)(10,14,13,15)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,39)(29,40)(31,32)(33,38)(34,35)(36,41)(37,43)(42,47)(44,45)(46,48);(1,4,5,2)(3,8,7,6)(17,20,22,32)(18,25,23,31)(19,28,30,41)(21,33,35,45)(24,36,27,39)(26,38,34,44)(29,42,43,48)(37,46,40,47);(17,21,22,35)(18,26,23,34)(19,29,30,43)(20,33,32,45)(24,37,27,40)(25,38,31,44)(28,42,41,48)(36,46,39,47)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022705);