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 | |
4114 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)(3,5)(4,12)(6,7)(8,13)(9,10)(11,16)(14,15)(18,22)(19,25)(20,28)(30,32)(33,34)(35,42)(36,39)(37,44)(38,40)(41,46)(43,47)(45,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,14)(12,16)(13,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,7)(3,9)(4,13)(14,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,13)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,6,13)(2,8,7,12)(3,11,9,15)(5,14,10,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,10)(2,9,7,3)(4,14,13,16)(8,15,12,11)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038416);