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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3867 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18,22,24)(19,23,27,30)(20,29,31,25)(21,26,32,28);(2,3)(7,9)(8,10)(15,16)(17,19,22,27)(18,23,24,30)(20,28,31,26)(21,25,32,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,3)(7,9)(8,10)(15,16)(18,24)(19,27)(25,29)(26,28)(34,39)(35,43)(41,45)(42,44);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,30)(25,32)(26,31)(33,35,38,43)(34,40,39,47)(36,44,46,42)(37,41,48,45);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20,22,31)(18,25,24,29)(19,28,27,26)(21,23,32,30)(33,36)(34,41)(35,44)(37,47)(38,46)(39,45)(40,48)(42,43);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,32)(18,26,24,28)(19,25,27,29)(20,30,31,23)(33,37,38,48)(34,42,39,44)(35,45,43,41)(36,40,46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=hi,e^2=h,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, 8632033);