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 | |
4050 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(36,44)(40,48)(42,45)(46,47);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,34,38,41)(35,37,43,39)(36,40,44,48)(42,46,45,47);(2,7)(3,10)(8,15)(11,12)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,35,38,43)(34,39,41,37)(36,42,44,45)(40,47,48,46);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,35)(34,37)(36,45)(38,43)(39,41)(40,47)(42,44)(46,48);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,37)(34,35)(36,46)(38,39)(40,42)(41,43)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297749);