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 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,18)(19,27)(20,30)(21,26)(22,24)(23,29)(25,31)(28,32);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19,22,29)(18,23,24,27)(20,28,31,26)(21,25,32,30)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(3,9)(6,13)(10,15)(14,16)(18,24)(20,31)(23,27)(26,28)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20)(18,25)(19,28)(21,27)(22,31)(23,32)(24,30)(26,29)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,21,22,32)(18,26,24,28)(19,25,29,30)(20,27,31,23)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442118);