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 | |
4279 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)(5,13,14,9)(11,15,16,12)(17,18,22,21)(19,25,26,23)(20,28,30,24)(27,31,29,32)(33,34,38,41)(35,36,39,43)(37,40,46,48)(42,44,45,47);(2,7)(4,8)(5,14)(11,16)(18,21)(23,25)(24,28)(31,32)(33,35)(34,36)(37,45)(38,39)(40,47)(41,43)(42,46)(44,48);(2,7)(3,10)(9,13)(11,16)(19,26)(23,25)(27,29)(31,32)(35,39)(36,43)(42,45)(44,47);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19)(18,23)(20,29)(21,25)(22,26)(24,32)(27,30)(28,31)(33,36,38,43)(34,39,41,35)(37,44,46,47)(40,45,48,42);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,30)(18,24,21,28)(19,27,26,29)(23,31,25,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21,22,18)(19,25,26,23)(20,28,30,24)(27,32,29,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,45)(41,48)(43,47)Polycyclic presentation:
2^9|a^2=gi,d^2=hi,e^2=g,f^2=g,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, 8746750);