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 | |
4181 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,28)(22,25)(23,27)(24,29)(26,32)(30,31)(33,34,38,42)(35,37,41,39)(36,44,46,40)(43,45,48,47);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,32)(28,30)(29,31)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(2,7)(3,8)(9,16)(11,13)(20,29)(24,28)(26,31)(30,32)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(1,3,6,8)(2,4,7,10)(5,13,15,11)(9,14,16,12)(17,20)(18,24)(19,26)(21,30)(22,29)(23,32)(25,28)(27,31)(33,35)(34,39)(36,43)(37,42)(38,41)(40,45)(44,47)(46,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,22,23)(18,19,25,27)(20,30,29,32)(24,26,28,31)(33,36,38,46)(34,40,42,44)(35,43,41,48)(37,47,39,45);(1,5,6,15)(2,9,7,16)(3,11,8,13)(4,12,10,14)(17,19)(18,21)(20,31)(22,27)(23,25)(24,32)(26,29)(28,30)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,43,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8628006);