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 | |
4092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,10)(8,11)(12,15)(13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(1,2,6,7)(3,11,10,8)(4,5,9,14)(12,16,15,13)(17,19)(18,23)(20,22)(21,27)(24,31)(25,26)(28,30)(29,32)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(2,7)(4,9)(8,11)(12,15)(17,18,22,25)(19,23,20,26)(21,24,30,29)(27,31,28,32)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,3,6,10)(2,8,7,11)(4,12,9,15)(5,13,14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,4)(2,5)(3,12)(6,9)(7,14)(8,13)(10,15)(11,16)(17,20,22,19)(18,23,25,26)(21,27,30,28)(24,32,29,31)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,16)(11,12)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9949784);