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 | |
4220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,39,43)(35,44,46,40)(36,37,47,41)(38,48,45,42);(3,10)(6,7)(11,16)(14,15)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,35)(34,40)(36,38)(37,42)(39,46)(41,48)(43,44)(45,47);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,39,47)(34,41,43,37)(35,38,46,45)(40,48,44,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36)(34,41)(35,45)(37,43)(38,46)(39,47)(40,42)(44,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,13)(14,16)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,37,39,41)(34,36,43,47)(35,42,46,48)(38,44,45,40);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,38,39,45)(34,42,43,48)(35,36,46,47)(37,44,41,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034821);