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 | |
4147 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,34,38,39)(35,40,43,37)(36,41,46,47)(42,48,44,45);(2,6)(4,13)(7,9)(11,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(34,39)(37,40)(41,47)(45,48);(2,6)(3,10)(8,12)(11,16)(18,22)(19,26)(24,28)(27,32)(33,35,38,43)(34,40,39,37)(36,44,46,42)(41,45,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36,38,46)(34,41,39,47)(35,42,43,44)(37,45,40,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37)(34,35)(36,45)(38,40)(39,43)(41,42)(44,47)(46,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8679642);