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 | |
4150 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)(3,7)(4,9)(5,13)(6,8)(10,14)(11,12)(15,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34)(35,46)(36,39)(37,44)(38,43)(40,45)(41,48)(42,47);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(18,24)(19,20)(26,29)(27,28)(33,35)(34,41)(36,37)(38,47)(39,43)(40,48)(42,44)(45,46);(2,8)(3,9)(10,13)(11,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,36,40,47)(34,42,45,39)(35,37,48,38)(41,44,46,43);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,37,40,38)(34,43,45,44)(35,36,48,47)(39,46,42,41);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,38,40,37)(34,44,45,43)(35,47,48,36)(39,41,42,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,39,40,42)(34,36,45,47)(35,43,48,44)(37,46,38,41)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8086864);