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 |
| prime radical | #2669 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4030 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)(3,10,5,7)(4,12,14,9)(11,15,13,16)(34,36)(39,42)(40,44)(46,48);(1,3,6,5)(2,7,8,10)(4,13,14,11)(9,16,12,15)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(35,41)(39,42)(43,47)(46,48);(2,8)(7,10)(9,12)(15,16)(18,23)(20,29)(24,31)(26,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,4,6,14)(2,9,8,12)(3,11,5,13)(7,15,10,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,2)(3,10)(4,9)(5,7)(6,8)(11,16)(12,14)(13,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,5,6,3)(2,7,8,10)(4,11,14,13)(9,16,12,15)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541091);