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 | |
4089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,13,7)(4,14,15,9)(5,10,16,12)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,3,6,13)(2,7,8,11)(4,10,15,12)(5,9,16,14)(19,20)(21,30)(23,26)(24,29)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(2,8)(5,16)(7,11)(9,14)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,11)(10,15)(14,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,4,6,15)(2,9,8,14)(3,12,13,10)(5,11,16,7)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,5)(2,10)(3,9)(4,11)(6,16)(7,15)(8,12)(13,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8154852);