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 | #58880 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,11,5)(4,12,13,8)(10,15,16,14)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3,6,11)(2,5,7,9)(4,10,13,16)(8,15,12,14)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(2,7)(4,13)(5,9)(10,16)(17,18,22,26)(19,24,20,23)(21,29,30,25)(27,31,28,32)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,4)(2,8)(3,10)(5,15)(6,13)(7,12)(9,14)(11,16)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,5)(2,3)(4,14)(6,9)(7,11)(8,16)(10,12)(13,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,2)(3,5)(4,8)(6,7)(9,11)(10,15)(12,13)(14,16)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900416);