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 | |
3904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,13,15)(4,10,11,5)(6,12,16,14)(33,34,40,44)(35,37,38,41)(36,45,46,42)(39,43,48,47);(1,3)(2,8)(4,14)(5,16)(6,10)(7,13)(9,15)(11,12)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,35)(34,41)(36,43)(37,44)(38,40)(39,45)(42,48)(46,47);(2,9)(3,13)(5,10)(12,14)(18,23)(20,29)(24,30)(26,32)(33,36,40,46)(34,42,44,45)(35,43,38,47)(37,39,41,48);(1,4,7,11)(2,10,9,5)(3,12,13,14)(6,8,16,15)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,37)(34,35)(36,39)(38,44)(40,41)(42,43)(45,47)(46,48);(1,5,7,10)(2,11,9,4)(3,6,13,16)(8,14,15,12)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,38,40,35)(34,37,44,41)(36,43,46,47)(39,42,48,45);(1,6)(2,12)(3,10)(4,15)(5,13)(7,16)(8,11)(9,14)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,39)(34,43)(35,45)(36,41)(37,46)(38,42)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^2=gi,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, 8525498);