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 | |
4107 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)(3,6)(4,9)(5,10)(7,12)(8,13)(11,14)(15,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,40)(36,38)(37,44)(39,42)(41,45)(43,46)(47,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(35,42)(39,40)(43,48)(46,47);(2,3)(7,9)(8,10)(14,15)(18,22)(23,25)(24,29)(30,31)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,34)(35,40,42,39)(37,44,45,41)(43,47,48,46);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,45)(34,41,36,44)(35,43,42,48)(39,46,40,47)Polycyclic presentation:
2^9|c^2=gh,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081813);