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 |
| prime radical | #40448 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4026 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,22,24)(19,23,26,20)(21,25,30,31)(27,29,32,28)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(2,7)(3,11)(4,14)(13,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,7)(4,14)(8,10)(12,16)(18,24)(19,26)(25,31)(27,32)(35,40)(37,45)(43,46)(44,48);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,20)(18,19)(21,29)(22,23)(24,26)(25,32)(27,31)(28,30)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,18)(19,20)(21,25)(22,24)(23,26)(27,28)(29,32)(30,31)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528541);