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 | |
4321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,4,7)(5,13,14,10)(11,15,12,16)(21,29)(24,30)(26,31)(27,32)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(2,8)(7,9)(10,13)(15,16)(18,23)(19,25)(24,30)(26,31)(33,35,38,36)(34,40,42,39)(37,43,46,44)(41,48,45,47);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,36,38,35)(34,39,42,40)(37,44,46,43)(41,47,45,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=g,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066600);