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 | |
4124 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18)(19,24)(20,30)(21,26)(22,23)(25,29)(27,32)(28,31)(33,34,39,44)(35,42,37,40)(36,41,47,45)(38,46,48,43);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(18,23)(21,30)(24,25)(28,32)(35,37)(38,48)(40,42)(43,46);(3,11)(4,13)(7,12)(8,16)(17,19,22,29)(18,24,23,25)(20,27,26,31)(21,28,30,32)(33,35,39,37)(34,40,44,42)(36,46,47,43)(38,45,48,41);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,19)(18,25)(20,31)(21,28)(22,29)(23,24)(26,27)(30,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,20,22,26)(18,21,23,30)(19,27,29,31)(24,28,25,32)(33,37,39,35)(34,42,44,40)(36,43,47,46)(38,41,48,45);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,21)(18,26)(19,28)(20,23)(22,30)(24,31)(25,27)(29,32)(33,38,39,48)(34,43,44,46)(35,45,37,41)(36,42,47,40)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=hi,e^2=gh,f^2=h,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, 8746341);