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 | |
3872 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)(3,6)(4,9)(5,10)(7,12)(8,13)(11,14)(15,16)(21,29)(24,30)(26,31)(27,32)(33,34,39,42)(35,43,46,40)(36,47,45,37)(38,44,48,41);(2,3)(7,9)(8,10)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,35,39,46)(34,40,42,43)(36,38,45,48)(37,44,47,41);(2,3)(7,9)(8,10)(14,15)(18,23)(19,25)(24,30)(26,31)(33,34,39,42)(35,40,46,43)(36,47,45,37)(38,41,48,44);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,36,39,45)(34,37,42,47)(35,38,46,48)(40,44,43,41);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,37)(34,36)(35,44)(38,40)(39,47)(41,46)(42,45)(43,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38)(34,41)(35,45)(36,46)(37,40)(39,48)(42,44)(43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,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, 8067000);