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 | |
4153 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)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,27)(20,29)(21,26)(22,24)(23,30)(25,31)(28,32)(33,34,39,41)(35,40,44,48)(36,38,43,46)(37,45,47,42);(2,6)(7,9)(8,13)(14,15)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(33,35,39,44)(34,40,41,48)(36,42,43,45)(37,46,47,38);(4,12)(8,13)(10,16)(14,15)(18,24)(21,32)(23,27)(25,29)(34,41)(35,44)(38,46)(42,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,23,32,27)(33,36,39,43)(34,38,41,46)(35,45,44,42)(37,40,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,37)(34,42)(35,38)(36,40)(39,47)(41,45)(43,48)(44,46);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,23,31,27)(33,38)(34,43)(35,37)(36,41)(39,46)(40,45)(42,48)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109393);