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 |
| prime radical | #58312 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3973 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)(3,4)(5,13)(6,7)(8,12)(9,14)(10,16)(11,15)(17,18,23,21)(19,26,22,24)(20,28,29,25)(27,31,30,32)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(1,3,6,8)(2,4,7,12)(5,10,14,15)(9,11,13,16)(17,19,23,22)(18,24,21,26)(20,27,29,30)(25,31,28,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(2,7)(3,8)(9,13)(10,15)(20,29)(25,28)(27,30)(31,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,20,23,29)(18,25,21,28)(19,27,22,30)(24,31,26,32)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,21,23,18)(19,26,22,24)(20,25,29,28)(27,31,30,32)(33,37,39,35)(34,36,42,40)(38,45,47,43)(41,44,46,48);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,22,23,19)(18,26,21,24)(20,30,29,27)(25,32,28,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=gh,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, 8066108);