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 | |
4244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18)(19,25)(20,28)(21,26)(22,23)(24,29)(27,31)(30,32)(35,41)(39,42)(43,47)(46,48);(2,6)(7,9)(8,12)(15,16)(18,23)(20,29)(21,25)(27,30)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(3,10)(7,9)(11,13)(15,16)(18,23)(19,26)(24,28)(27,30)(34,38)(39,42)(40,44)(46,48);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,19)(18,21)(20,30)(22,26)(23,25)(24,32)(27,29)(28,31)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(17,20)(18,24)(19,27)(21,31)(22,29)(23,28)(25,32)(26,30)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21)(18,19)(20,31)(22,25)(23,26)(24,27)(28,30)(29,32)(33,36,37,45)(34,40,38,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8783525);