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 | |
4185 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)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(1,3,7,9)(2,4,5,12)(6,10,15,16)(8,11,13,14)(17,19)(18,24)(20,28)(21,26)(22,29)(23,27)(25,32)(30,31)(34,40)(35,36)(42,47)(43,44);(2,5)(3,9)(8,13)(10,16)(19,27)(21,24)(28,32)(29,30)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,4)(2,3)(5,9)(6,14)(7,12)(8,16)(10,13)(11,15)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,5,7,2)(3,4,9,12)(6,13,15,8)(10,11,16,14)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,6)(2,8)(3,10)(4,11)(5,13)(7,15)(9,16)(12,14)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8495594);