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 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,10,12)(4,8,11,13)(9,14,15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,31)(29,30);(2,6)(7,12)(8,13)(14,16)(17,19,23,20)(18,24,25,26)(21,28,32,30)(22,29,27,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(4,11)(8,13)(9,15)(14,16)(18,25)(22,27)(24,26)(29,31)(34,38)(39,41)(40,45)(46,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20,23,19)(18,26,25,24)(21,30,32,28)(22,31,27,29)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,21)(18,27)(19,28)(20,30)(22,25)(23,32)(24,31)(26,29)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,4,5,11)(2,8,6,13)(3,9,10,15)(7,14,12,16)(17,22,23,27)(18,21,25,32)(19,29,20,31)(24,28,26,30)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8100941);