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 | |
4045 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)(8,9)(10,14)(11,15)(12,16)(18,23)(19,27)(21,25)(28,30)(33,34)(35,39)(36,44)(37,42)(38,41)(40,46)(43,48)(45,47);(2,7)(3,9)(10,13)(11,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,35,38,37)(34,39,41,42)(36,43,46,45)(40,47,44,48);(2,7)(3,9)(10,13)(11,16)(19,27)(24,31)(28,30)(29,32)(33,34,38,41)(35,42,37,39)(36,40,46,44)(43,48,45,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19)(18,24)(20,30)(21,29)(22,27)(23,31)(25,32)(26,28)(33,34)(35,42)(36,44)(37,39)(38,41)(40,46)(43,47)(45,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,22,26)(18,25,23,21)(19,28,27,30)(24,29,31,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,22,25)(18,26,23,20)(19,29,27,32)(24,28,31,30)(33,37,38,35)(34,39,41,42)(36,45,46,43)(40,47,44,48)Polycyclic presentation:
2^9|b^2=i,c^2=i,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, 8065893);