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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4099 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)(11,16)(13,14)(18,23)(20,24)(26,29)(28,31)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,5,6,7)(2,9,10,3)(4,13,12,14)(8,16,15,11)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,e^2=ghi,f^2=i,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, 8366009);