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 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,12)(10,16)(14,15)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,48)(44,47)(45,46);(2,4)(7,9)(8,12)(14,15)(18,23)(19,26)(25,28)(27,32)(34,39)(40,46)(41,42)(47,48);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,35,38,45)(34,40,39,46)(36,43,37,44)(41,47,42,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,36,38,37)(34,41,39,42)(35,43,45,44)(40,47,46,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,37,38,36)(34,42,39,41)(35,44,45,43)(40,48,46,47)Polycyclic presentation:
2^9|c^2=ghi,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017679);