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 | |
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:
(1,2)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18)(19,24)(20,25)(21,31)(22,27)(23,28)(26,30)(29,32)(33,34,38,42)(35,39,45,48)(36,37,41,40)(43,47,46,44);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,31)(33,35,38,45)(34,39,42,48)(36,46,41,43)(37,44,40,47);(17,19,23,32)(18,24,28,29)(20,30,27,21)(22,31,25,26)(33,36,38,41)(34,40,42,37)(35,43,45,46)(39,44,48,47);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,20)(18,25)(19,30)(21,32)(22,28)(23,27)(24,26)(29,31)(33,35)(34,39)(36,43)(37,47)(38,45)(40,44)(41,46)(42,48);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,21,23,30)(18,26,28,31)(19,20,32,27)(22,29,25,24);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,22,23,25)(18,27,28,20)(19,31,32,26)(21,29,30,24)(33,37,38,40)(34,41,42,36)(35,44,45,47)(39,43,48,46)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=ghi,f^2=gi,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, 8371585);