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 | |
3882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,8,4)(5,9,14,13)(11,15,16,12)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(35,43)(36,45)(39,46)(40,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,16)(13,14)(18,23)(19,26)(25,28)(27,32)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(1,3,6,8)(2,7,10,4)(5,11,14,16)(9,15,13,12)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46);(1,2)(3,7)(4,8)(5,13)(6,10)(9,14)(11,12)(15,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,16,13,11)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,45)(34,40,42,48)(35,44,43,37)(39,47,46,41);(1,5,6,14)(2,9,10,13)(3,11,8,16)(4,12,7,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,37,38,44)(34,41,42,47)(35,45,43,36)(39,48,46,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=g,e^2=ghi,f^2=ghi,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, 8370878);