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 | |
4261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,18)(19,24)(20,23)(21,25)(22,28)(26,30)(27,29)(31,32)(35,41)(39,46)(42,44)(47,48);(2,4)(7,11)(8,12)(14,16)(17,19)(18,24)(20,21)(22,27)(23,25)(26,31)(28,29)(30,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(3,9)(5,13)(7,11)(8,12)(18,20)(19,25)(22,30)(29,31)(34,38)(35,41)(40,43)(42,44);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,20,23,18)(19,21,25,24)(22,28,30,26)(27,29,32,31)(33,35,37,41)(34,39,38,46)(36,44,45,42)(40,48,43,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21)(18,25)(19,20)(22,29)(23,24)(26,32)(27,28)(30,31)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,30)(18,26,20,28)(19,27,25,32)(21,29,24,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=g,d^2=gi,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=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837550);