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 | |
3993 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)(3,10)(4,9)(5,14)(6,7)(8,11)(12,16)(13,15)(17,18,22,26)(19,23,27,31)(20,24,30,28)(21,25,32,29)(34,39)(35,42)(37,45)(44,47);(2,7)(3,11)(4,14)(13,16)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(3,11)(8,10)(12,15)(13,16)(19,27)(20,30)(23,31)(24,28)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20)(18,24)(19,25)(21,31)(22,30)(23,32)(26,28)(27,29)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,14)(2,9,7,5)(3,12,11,15)(8,13,10,16)(17,21)(18,25)(19,28)(20,23)(22,32)(24,27)(26,29)(30,31)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,18)(19,23)(20,24)(21,29)(22,26)(25,32)(27,31)(28,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10219580);