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:
(1,2,6,10)(3,11,13,7)(4,8,15,12)(5,9,16,14)(17,18)(19,24)(20,27)(21,29)(22,23)(25,31)(26,28)(30,32)(33,34)(35,37)(36,39)(38,44)(40,42)(41,46)(43,47)(45,48);(1,3,6,13)(2,7,10,11)(4,9,15,14)(5,12,16,8)(17,19,22,27)(18,20,23,24)(21,26,31,30)(25,28,29,32)(33,35)(34,40)(36,37)(38,43)(39,42)(41,47)(44,45)(46,48);(18,23)(20,24)(25,29)(28,32)(34,36)(35,42)(41,44)(43,48);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,15)(10,11)(17,20)(18,19)(21,28)(22,24)(23,27)(25,26)(29,30)(31,32)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4)(2,8)(3,9)(5,7)(6,15)(10,12)(11,16)(13,14)(17,19)(18,24)(20,23)(21,30)(22,27)(25,28)(26,31)(29,32)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5)(2,9)(3,12)(4,11)(6,16)(7,15)(8,13)(10,14)(17,21,22,31)(18,25,23,29)(19,26,27,30)(20,28,24,32)(33,38,39,46)(34,41,36,44)(35,43,42,48)(37,45,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=i,f^2=h,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, 10154589);