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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,21)(20,28)(22,24)(23,26)(25,31)(27,32)(29,30)(33,34,40,43)(35,37,44,36)(38,41,48,46)(39,42,47,45);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,15)(14,16)(17,19,22,26)(18,23,24,21)(20,29,31,27)(25,30,28,32)(33,35,40,44)(34,36,43,37)(38,47,48,39)(41,42,46,45);(2,6)(4,13)(7,9)(10,14)(18,24)(21,23)(25,28)(30,32)(33,36,40,37)(34,35,43,44)(38,42,48,45)(39,41,47,46);(1,3)(2,7)(4,14)(5,11)(6,9)(8,16)(10,13)(12,15)(17,20,22,31)(18,25,24,28)(19,27,26,29)(21,30,23,32)(33,37,40,36)(34,35,43,44)(38,45,48,42)(39,41,47,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,18)(19,23)(20,28)(21,26)(22,24)(25,31)(27,30)(29,32)(33,38)(34,41)(35,39)(36,45)(37,42)(40,48)(43,46)(44,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,13)(14,16)(17,21,22,23)(18,26,24,19)(20,30,31,32)(25,29,28,27)(33,39)(34,42)(35,38)(36,46)(37,41)(40,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,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, 8948116);