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 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,4)(10,14)(11,12)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34,39,38)(35,40,45,44)(36,46,41,37)(42,48,47,43);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(18,23)(19,26)(25,28)(27,32)(33,35,39,45)(34,40,38,44)(36,42,41,47)(37,43,46,48);(1,3,6,4)(2,8,7,9)(5,12,13,11)(10,16,14,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,36)(34,37)(35,42)(38,46)(39,41)(40,43)(44,48)(45,47);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35)(34,40)(36,42)(37,43)(38,44)(39,45)(41,47)(46,48);(1,4,6,3)(2,9,7,8)(5,12,13,11)(10,16,14,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,46)(34,41,38,36)(35,43,45,48)(40,47,44,42);(1,5,6,13)(2,10,7,14)(3,11,4,12)(8,15,9,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,38,39,34)(35,44,45,40)(36,46,41,37)(42,48,47,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,e^2=h,f^2=h,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, 8087656);