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 | |
3968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,9,14,15)(11,12,13,16)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(2,7)(3,10)(9,15)(11,13)(17,19,23,27)(18,24,26,21)(20,28,25,32)(22,29,31,30)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(2,7)(4,8)(9,15)(12,16)(19,27)(21,24)(28,32)(29,30)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,14)(15,16)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,4,6,8)(2,3,7,10)(5,12,14,16)(9,11,15,13)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,5,6,14)(2,9,7,15)(3,11,10,13)(4,12,8,16)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,e^2=g,f^2=g,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, 8428810);