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 | |
4099 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,22,23)(19,25,30,24)(20,26,27,21)(28,29,31,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(18,23)(20,27)(24,25)(28,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(4,12)(8,15)(11,13)(14,16)(17,19,22,30)(18,24,23,25)(20,28,27,31)(21,29,26,32)(33,35,37,43)(34,38,41,39)(36,42,45,48)(40,46,44,47);(1,4,6,12)(2,8,9,15)(3,11,5,13)(7,14,10,16)(17,19)(18,25)(20,31)(21,29)(22,30)(23,24)(26,32)(27,28)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,20)(18,26)(19,28)(21,23)(22,27)(24,32)(25,29)(30,31)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21,22,26)(18,27,23,20)(19,29,30,32)(24,31,25,28)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091086);