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 | |
3970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18,22,26)(19,27,29,23)(20,24,30,32)(21,25,31,28)(33,34,39,43)(35,44,45,40)(36,41,47,46)(37,38,42,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,22,29)(18,23,26,27)(20,28,30,25)(21,24,31,32)(35,45)(36,47)(40,44)(41,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32)(33,35,39,45)(34,40,43,44)(36,42,47,37)(38,46,48,41);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,27)(20,24)(21,28)(22,26)(23,29)(25,31)(30,32)(33,36)(34,41)(35,42)(37,45)(38,40)(39,47)(43,46)(44,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20,22,30)(18,24,26,32)(19,28,29,25)(21,23,31,27)(33,37,39,42)(34,38,43,48)(35,36,45,47)(40,46,44,41);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,21)(18,25)(19,24)(20,23)(22,31)(26,28)(27,30)(29,32)(33,38)(34,42)(35,46)(36,44)(37,43)(39,48)(40,47)(41,45)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,e^2=h,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, 8338123);