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 | |
4171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,14)(10,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(37,45)(41,46)(43,44)(47,48);(1,2,5,6)(3,7,9,13)(4,8,12,14)(10,15,11,16)(17,19,22,25)(18,23,26,21)(20,28,30,27)(24,29,32,31)(33,34,39,38)(35,42,36,40)(37,46,45,41)(43,47,44,48);(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,30)(31,32)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(1,3)(2,7)(4,11)(5,9)(6,13)(8,16)(10,12)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36,39,35)(34,40,38,42)(37,44,45,43)(41,47,46,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,29,23,31)(33,37,39,45)(34,41,38,46)(35,43,36,44)(40,47,42,48);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,28,32,27)(33,38,39,34)(35,42,36,40)(37,46,45,41)(43,48,44,47)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416154);