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 | |
4109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18)(19,24)(20,26)(21,27)(22,29)(23,25)(28,32)(30,31)(33,34)(35,43)(36,45)(37,42)(38,40)(39,46)(41,47)(44,48);(2,6)(4,12)(7,10)(11,15)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,26,32,30)(33,35,38,46)(34,39,40,43)(36,42,47,44)(37,45,48,41);(3,9)(7,10)(11,15)(14,16)(18,25)(19,21)(26,30)(28,29)(34,40)(37,48)(39,43)(41,45);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,27,32,24)(33,36,38,47)(34,41,40,45)(35,44,46,42)(37,39,48,43);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,21,23,19)(18,27,25,24)(20,28,31,29)(22,26,32,30)(33,37,38,48)(34,42,40,44)(35,45,46,41)(36,39,47,43);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,27,31,24)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10235300);