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 | |
4055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,22,20)(19,23,26,27)(21,24,29,28)(25,30,31,32)(35,41)(39,46)(42,44)(47,48);(4,11)(7,9)(12,16)(14,15)(18,20)(23,27)(24,28)(30,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,3)(7,9)(8,10)(14,15)(21,29)(24,28)(25,31)(30,32)(34,38)(39,46)(40,43)(47,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,35,37,41)(34,39,38,46)(36,44,45,42)(40,48,43,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,31,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21)(18,24)(19,25)(20,28)(22,29)(23,30)(26,31)(27,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388618);