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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3993 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,26)(19,23,27,31)(20,24,30,28)(21,25,32,29)(33,34)(35,45)(36,41)(37,43)(38,48)(39,44)(40,46)(42,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,35)(34,40)(36,38)(37,41)(39,46)(42,48)(43,47)(44,45);(2,3)(7,9)(8,10)(14,15)(19,27)(20,30)(23,31)(24,28)(33,36,39,47)(34,41,44,42)(35,38,46,43)(37,45,48,40);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,24)(19,25)(21,31)(22,30)(23,32)(26,28)(27,29)(33,36)(34,42)(35,43)(37,40)(38,46)(39,47)(41,44)(45,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21)(18,25)(19,28)(20,23)(22,32)(24,27)(26,29)(30,31)(33,37,39,48)(34,43,44,38)(35,42,46,41)(36,45,47,40);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,32)(27,31)(28,30)(33,38,39,43)(34,37,44,48)(35,36,46,47)(40,41,45,42)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219578);