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 | #2813 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(3,9)(8,12)(11,13)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(33,34,37,38)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(19,29)(20,31)(24,28)(25,30)(34,38)(39,41)(40,44)(46,47);(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,5,6,14)(2,8,4,12)(3,11,9,13)(7,15,10,16)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8344028);