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 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,9,4)(5,13,15,10)(11,12,14,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(34,38)(35,41)(36,43)(47,48);(1,3)(2,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(33,34,37,38)(35,39,41,46)(36,44,43,40)(42,48,45,47);(2,8)(3,9)(10,13)(11,14)(19,27)(20,29)(23,28)(24,32)(33,35,37,41)(34,39,38,46)(36,42,43,45)(40,47,44,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,16)(12,13)(14,15)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,35)(34,39)(36,45)(37,41)(38,46)(40,48)(42,43)(44,47);(1,4,6,7)(2,9,8,3)(5,12,15,16)(10,14,13,11)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,36)(34,40)(35,42)(37,43)(38,44)(39,47)(41,45)(46,48);(1,5,6,15)(2,10,8,13)(3,11,9,14)(4,12,7,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,44)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=ghi,e^2=gh,f^2=gh,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, 8746284);