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 | |
4144 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,13)(14,16)(17,18)(19,27)(20,30)(21,24)(22,28)(23,25)(26,31)(29,32)(33,34,39,41)(35,40,43,37)(36,42,46,48)(38,44,45,47);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,30,32,26)(33,35,39,43)(34,40,41,37)(36,45,46,38)(42,47,48,44);(5,13)(9,12)(10,16)(14,15)(18,25)(22,32)(24,27)(26,30)(34,41)(35,43)(38,45)(42,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27)(33,36)(34,42)(35,38)(37,47)(39,46)(40,44)(41,48)(43,45);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21,23,19)(18,27,25,24)(20,29,31,28)(22,26,32,30)(33,37,39,40)(34,43,41,35)(36,44,46,47)(38,48,45,42);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,24,31,27)(33,38)(34,44)(35,36)(37,48)(39,45)(40,42)(41,47)(43,46)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231660);