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 | |
4274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,5)(6,10)(7,13)(8,14)(11,16)(12,15)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,16)(9,10)(14,15)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(34,39)(36,42)(41,45)(44,48);(4,14)(5,8)(11,15)(12,16)(18,23)(19,26)(21,30)(27,29)(33,35,38,40)(34,36,39,42)(37,46,47,43)(41,48,45,44);(1,4)(2,8)(3,11)(5,10)(6,14)(7,16)(9,12)(13,15)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,3)(2,9)(4,15)(5,12)(6,13)(7,10)(8,16)(11,14)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,36)(34,40)(35,39)(37,44)(38,42)(41,46)(43,45)(47,48);(1,5,6,8)(2,4,10,14)(3,12,13,16)(7,11,9,15)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,37,38,47)(34,41,39,45)(35,43,40,46)(36,44,42,48)Polycyclic presentation:
2^9|b^2=i,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8825031);