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 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,22,24)(19,23,28,25)(20,26,31,29)(21,30,32,27)(33,34,38,39)(35,41,46,40)(36,42,47,45)(37,44,48,43);(4,11)(8,14)(10,15)(12,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,25)(26,32)(27,31)(34,39)(36,47)(40,41)(43,44);(2,7)(3,9)(8,14)(10,15)(18,24)(19,28)(26,29)(27,30)(33,35,38,46)(34,40,39,41)(36,44,47,43)(37,45,48,42);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,25)(20,27)(21,29)(22,28)(23,24)(26,32)(30,31)(33,35)(34,41)(36,44)(37,45)(38,46)(39,40)(42,48)(43,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,31)(18,26,24,29)(19,27,28,30)(21,25,32,23)(33,36)(34,42)(35,44)(37,40)(38,47)(39,45)(41,48)(43,46);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21,22,32)(18,27,24,30)(19,29,28,26)(20,23,31,25)(33,37,38,48)(34,43,39,44)(35,45,46,42)(36,41,47,40)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619609);