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 | |
4118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,10,11,8)(4,9,14,5)(12,16,15,13)(17,18)(19,24)(20,22)(21,26)(23,25)(27,31)(28,32)(29,30);(2,7)(3,11)(4,14)(13,16)(17,19,23,28)(18,24,25,32)(20,30,27,21)(22,29,31,26)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(4,14)(5,9)(12,15)(13,16)(18,25)(19,28)(21,30)(22,31)(34,38)(36,45)(39,41)(42,46);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20,23,27)(18,22,25,31)(19,21,28,30)(24,26,32,29)(33,35)(34,39)(36,46)(37,43)(38,41)(40,48)(42,45)(44,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,21)(18,26)(19,20)(22,24)(23,30)(25,29)(27,28)(31,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22)(18,27)(19,29)(20,25)(21,24)(23,31)(26,28)(30,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947700);