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 | |
4027 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,34,38,44)(35,42,46,39)(36,41,47,40)(37,45,48,43);(2,7)(4,8)(10,15)(12,13)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,35)(34,39)(36,45)(37,41)(38,46)(40,48)(42,44)(43,47);(18,23)(19,25)(24,31)(26,32)(33,36,38,47)(34,40,44,41)(35,45,46,43)(37,39,48,42);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,32)(28,31)(33,36)(34,41)(35,45)(37,39)(38,47)(40,44)(42,48)(43,46);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,20,22,28)(18,24,23,31)(19,26,25,32)(21,29,30,27)(33,35)(34,42)(36,45)(37,40)(38,46)(39,44)(41,48)(43,47);(1,5,6,14)(2,10,7,15)(3,11,9,16)(4,12,8,13)(17,21)(18,19)(20,27)(22,30)(23,25)(24,32)(26,31)(28,29)(33,37,38,48)(34,43,44,45)(35,41,46,40)(36,39,47,42)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9830839);