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 | |
3945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,9,4)(5,13,15,10)(11,12,14,16)(17,18,23,28)(19,26,21,24)(20,30,29,25)(22,27,31,32)(33,34)(35,41)(36,43)(37,45)(38,40)(39,47)(42,44)(46,48);(1,3)(2,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(17,19,23,21)(18,24,28,26)(20,31,29,22)(25,32,30,27)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(2,8)(3,9)(10,13)(11,14)(17,20)(18,25)(19,22)(21,31)(23,29)(24,27)(26,32)(28,30)(33,34,38,40)(35,36,43,41)(37,39,47,45)(42,44,46,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,16)(12,13)(14,15)(17,21,23,19)(18,26,28,24)(20,31,29,22)(25,32,30,27)(33,36,38,41)(34,35,40,43)(37,44,47,48)(39,42,45,46);(1,4,6,7)(2,9,8,3)(5,12,15,16)(10,14,13,11)(17,22,23,31)(18,27,28,32)(19,29,21,20)(24,30,26,25)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,48)(46,47);(1,5,6,15)(2,10,8,13)(3,11,9,14)(4,12,7,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,37,38,47)(34,39,40,45)(35,42,43,46)(36,44,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,d^2=g,e^2=i,f^2=gi,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, 8796575);