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 | #45020 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,29)(27,31)(28,32)(33,34,40,38)(35,39,43,41)(36,42,37,47)(44,46,45,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,28)(31,32)(33,35,40,43)(34,41,38,39)(36,44,37,45)(42,48,47,46);(2,5)(8,10)(9,12)(15,16)(18,23)(20,30)(24,26)(27,32)(33,36,40,37)(34,42,38,47)(35,44,43,45)(39,46,41,48);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,19)(18,24)(20,27)(21,31)(22,29)(23,26)(25,28)(30,32)(33,37,40,36)(34,42,38,47)(35,45,43,44)(39,46,41,48);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,31)(29,32)(33,38,40,34)(35,41,43,39)(36,42,37,47)(44,46,45,48);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,32,26,27)(33,39,40,41)(34,43,38,35)(36,46,37,48)(42,45,47,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=gi,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, 8546939);