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 | |
4058 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18)(19,28)(20,26)(21,22)(23,25)(24,27)(29,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19,24,23)(18,25,27,28)(20,29,21,30)(22,31,26,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(2,8)(3,11)(4,13)(10,14)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,32,28,31)(33,35,37,43)(34,38,40,41)(36,42,46,45)(39,47,44,48);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,21,24,20)(18,26,27,22)(19,30,23,29)(25,32,28,31)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,22,24,26)(18,21,27,20)(19,31,23,32)(25,30,28,29)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=ghi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10109236);