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 | |
4046 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)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,3)(2,5)(4,11)(6,9)(7,14)(8,10)(12,16)(13,15)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(4,12)(7,15)(11,16)(13,14)(19,27)(20,28)(23,32)(24,29)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,4,6,12)(2,7,10,15)(3,11,9,16)(5,14,8,13)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,3)(2,8)(4,11)(5,10)(6,9)(7,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,5,6,8)(2,9,10,3)(4,13,12,14)(7,11,15,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8642118);