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 | |
4134 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)(4,5,9,15)(12,13,14,16)(17,18)(19,24)(20,26)(21,27)(22,31)(23,25)(28,30)(29,32)(33,34)(35,42)(36,44)(37,41)(38,39)(40,46)(43,48)(45,47);(2,7)(3,11)(4,9)(13,16)(17,19,23,22)(18,24,25,31)(20,28,29,27)(21,26,30,32)(33,35)(34,37)(36,45)(38,42)(39,41)(40,47)(43,46)(44,48);(3,11)(4,9)(5,15)(8,10)(18,25)(19,22)(20,29)(21,30)(34,38)(36,46)(37,42)(43,45);(1,3,6,11)(2,8,7,10)(4,14,9,12)(5,13,15,16)(17,20,23,29)(18,26,25,32)(19,27,22,28)(21,31,30,24)(33,36)(34,40)(35,43)(37,48)(38,44)(39,46)(41,45)(42,47);(1,4,6,9)(2,5,7,15)(3,12,11,14)(8,16,10,13)(17,21)(18,27)(19,26)(20,24)(22,32)(23,30)(25,28)(29,31)(33,37,39,42)(34,41,38,35)(36,47,46,48)(40,43,44,45);(1,5)(2,9)(3,13)(4,7)(6,15)(8,12)(10,14)(11,16)(17,22,23,19)(18,24,25,31)(20,28,29,27)(21,32,30,26)(33,38,39,34)(35,42,41,37)(36,44,46,40)(43,47,45,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8969336);