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 | |
4185 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(33,34)(35,43)(36,41)(37,47)(38,39)(40,45)(42,48)(44,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(34,39)(36,46)(40,43)(42,47);(2,6)(3,5)(4,12)(15,16)(18,23)(19,27)(20,30)(21,31)(33,35,38,45)(34,40,39,43)(36,47,46,42)(37,41,48,44);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,35)(34,40)(36,47)(37,44)(38,45)(39,43)(41,48)(42,46);(1,5,7,3)(2,10,6,8)(4,11,12,14)(9,15,13,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,36,38,46)(34,41,39,44)(35,42,45,47)(37,43,48,40);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,37)(34,42)(35,44)(36,43)(38,48)(39,47)(40,46)(41,45)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8886910);