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 | |
4021 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)(5,13)(6,7)(8,15)(9,10)(11,14)(12,16)(17,18)(19,27)(20,25)(21,31)(22,23)(24,26)(28,32)(29,30)(34,39)(35,43)(37,46)(44,47);(1,3)(2,4)(5,11)(6,9)(7,10)(8,12)(13,14)(15,16)(17,19,24,22)(18,23,26,27)(20,28,21,29)(25,30,31,32)(33,34)(35,40)(36,46)(37,42)(38,39)(41,43)(44,48)(45,47);(2,7)(3,9)(8,13)(11,16)(17,20,24,21)(18,25,26,31)(19,28,22,29)(23,30,27,32)(33,35,38,43)(34,40,39,41)(36,44,42,47)(37,45,46,48);(1,2)(3,10)(4,9)(5,8)(6,7)(11,14)(12,16)(13,15)(17,21,24,20)(18,25,26,31)(19,28,22,29)(23,32,27,30)(33,35)(34,41)(36,47)(37,45)(38,43)(39,40)(42,44)(46,48);(1,4,6,10)(2,3,7,9)(5,14,15,12)(8,16,13,11)(17,22,24,19)(18,23,26,27)(20,29,21,28)(25,30,31,32)(33,36)(34,37)(35,44)(38,42)(39,46)(40,45)(41,48)(43,47);(1,5)(2,8)(3,11)(4,12)(6,15)(7,13)(9,16)(10,14)(17,23)(18,22)(19,26)(20,30)(21,32)(24,27)(25,29)(28,31)(33,37)(34,42)(35,45)(36,39)(38,46)(40,47)(41,44)(43,48)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154346);