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 | |
3979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(2,4)(3,9)(8,12)(10,15)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(5,13)(8,12)(10,15)(14,16)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313655);