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:
(2,7)(3,9)(8,15)(11,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,40,44)(35,45,47,41)(36,48,46,42)(37,39,38,43);(1,2,6,7)(3,10,9,5)(4,8,13,15)(11,12,16,14)(19,28)(20,29)(24,30)(25,32)(33,35)(34,41)(36,37)(38,46)(39,42)(40,47)(43,48)(44,45);(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,36,40,46)(34,42,44,48)(35,37,47,38)(39,45,43,41);(1,3,6,9)(2,5,7,10)(4,11,13,16)(8,14,15,12)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,37,40,38)(34,43,44,39)(35,36,47,46)(41,48,45,42);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,16)(10,12)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,38,40,37)(34,39,44,43)(35,46,47,36)(41,42,45,48);(1,5)(2,3)(4,12)(6,10)(7,9)(8,16)(11,15)(13,14)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,39)(34,38)(35,42)(36,45)(37,44)(40,43)(41,46)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8603408);