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 | |
4463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,4)(6,7)(10,12)(14,15)(17,18,23,24)(19,21,30,25)(20,31,32,26)(22,37,38,27)(28,39,33,34)(29,40,43,36)(35,41,45,44)(42,47,46,48);(1,2,5,8)(3,6,4,7)(9,10,13,12)(11,15,16,14)(18,24)(20,32)(21,25)(27,37)(28,33)(35,45)(36,40)(42,46);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(17,19)(18,25)(20,33)(21,24)(22,29)(23,30)(26,34)(27,40)(28,32)(31,39)(35,46)(36,37)(38,43)(41,47)(42,45)(44,48);(1,2)(3,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,23,32)(18,26,24,31)(19,28,30,33)(21,34,25,39)(22,35,38,45)(27,41,37,44)(29,42,43,46)(36,47,40,48);(1,4,5,3)(2,7,8,6)(17,21,23,25)(18,19,24,30)(20,34,32,39)(22,36,38,40)(26,28,31,33)(27,29,37,43)(35,47,45,48)(41,42,44,46);(17,22,23,38)(18,27,24,37)(19,29,30,43)(20,35,32,45)(21,36,25,40)(26,41,31,44)(28,42,33,46)(34,47,39,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,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, 8022676);