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 | #53841 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(4,12)(14,16)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(34,38)(39,41)(40,45)(46,47);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021588);