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 | |
4176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,9,14,15)(11,12,13,16)(18,23)(19,25)(24,30)(26,31);(2,7)(3,10)(9,15)(11,13)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(2,7)(4,8)(9,15)(12,16)(20,27)(24,30)(26,31)(28,32)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,14)(15,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,4,6,8)(2,3,7,10)(5,12,14,16)(9,11,15,13)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442071);