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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,4,5,8)(6,14,15,10)(11,16,13,12)(21,29)(24,30)(26,31)(27,32)(33,34)(35,44)(36,37)(38,47)(39,43)(40,46)(41,45)(42,48);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,16,14,12)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,35,39,46)(34,40,43,44)(36,42,41,47)(37,38,45,48);(2,9)(4,8)(10,14)(12,16)(18,23)(19,25)(24,30)(26,31)(33,36,39,41)(34,37,43,45)(35,42,46,47)(38,44,48,40);(1,4,7,8)(2,3,9,5)(6,12,15,16)(10,11,14,13)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,37,39,45)(34,41,43,36)(35,38,46,48)(40,47,44,42);(1,5,7,3)(2,4,9,8)(6,13,15,11)(10,12,14,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,34)(35,40)(36,37)(38,42)(39,43)(41,45)(44,46)(47,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38)(34,42)(35,45)(36,44)(37,46)(39,48)(40,41)(43,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066919);