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 | |
4270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(18,23)(19,25)(24,31)(26,28)(33,34,38,39)(35,43,46,40)(36,41,47,45)(37,44,48,42);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,31)(28,32)(34,39)(36,47)(40,43)(42,44);(2,6)(3,9)(4,12)(14,16)(19,25)(20,27)(21,30)(24,31)(33,35,38,46)(34,40,39,43)(36,42,47,44)(37,41,48,45);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,21)(20,28)(22,25)(23,30)(24,32)(26,27)(29,31)(33,35)(34,40)(36,42)(37,45)(38,46)(39,43)(41,48)(44,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36)(34,41)(35,44)(37,40)(38,47)(39,45)(42,46)(43,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21)(18,19)(20,29)(22,30)(23,25)(24,26)(27,32)(28,31)(33,37)(34,42)(35,45)(36,40)(38,48)(39,44)(41,46)(43,47)Polycyclic presentation:
2^9|a^2=h,c^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8925268);