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 | #48138 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(4,12)(7,10)(8,13)(17,18,22,23)(19,21,25,32)(20,27,29,24)(26,30,28,31)(33,34,38,43)(35,42,37,39)(36,41,46,40)(44,47,45,48);(1,2)(3,10)(4,13)(5,6)(7,9)(8,12)(11,15)(14,16)(17,19)(18,21)(20,28)(22,25)(23,32)(24,30)(26,29)(27,31)(33,35,38,37)(34,39,43,42)(36,44,46,45)(40,47,41,48);(2,6)(3,9)(8,13)(11,14)(18,23)(19,25)(20,29)(30,31)(33,36,38,46)(34,40,43,41)(35,44,37,45)(39,47,42,48);(1,3)(2,7)(4,14)(5,9)(6,10)(8,16)(11,12)(13,15)(17,20,22,29)(18,24,23,27)(19,26,25,28)(21,31,32,30)(33,36)(34,41)(35,45)(37,44)(38,46)(39,47)(40,43)(42,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,15)(9,14)(10,16)(17,21)(18,19)(20,30)(22,32)(23,25)(24,28)(26,27)(29,31)(33,37,38,35)(34,42,43,39)(36,45,46,44)(40,48,41,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,24)(22,23)(25,32)(26,31)(27,29)(28,30)(33,34)(35,39)(36,41)(37,42)(38,43)(40,46)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840325);