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 | |
4123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(34,39)(35,44)(36,43)(46,47);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,14)(13,16)(17,18,22,23)(19,24,26,20)(21,25,31,32)(27,30,29,28)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,47)(46,48);(4,13)(7,12)(11,16)(14,15)(18,23)(19,26)(25,32)(27,29)(33,35,38,44)(34,40,39,41)(36,45,43,48)(37,46,42,47);(1,4,6,13)(2,7,10,12)(3,11,9,16)(5,15,8,14)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,31)(30,32)(33,35)(34,41)(36,45)(37,47)(38,44)(39,40)(42,46)(43,48);(1,3)(2,8)(4,11)(5,10)(6,9)(7,14)(12,15)(13,16)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,29,32,27)(33,36)(34,42)(35,45)(37,39)(38,43)(40,47)(41,46)(44,48);(1,5,6,8)(2,9,10,3)(4,14,13,15)(7,11,12,16)(17,21,22,31)(18,25,23,32)(19,27,26,29)(20,28,24,30)(33,37,38,42)(34,43,39,36)(35,46,44,47)(40,48,41,45)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633458);