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 | |
3975 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)(3,11)(4,9)(5,10)(6,8)(7,14)(12,16)(13,15)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34,39,43)(35,42,45,38)(36,40,37,41)(44,46,47,48);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,13,16,9)(18,23)(19,26)(25,28)(27,32)(33,35,39,45)(34,38,43,42)(36,44,37,47)(40,48,41,46);(2,8)(5,16)(7,11)(9,13)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,36,39,37)(34,40,43,41)(35,44,45,47)(38,48,42,46);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,11,16,7)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,37,39,36)(34,41,43,40)(35,44,45,47)(38,48,42,46);(1,3)(2,7)(4,10)(5,13)(6,14)(8,11)(9,16)(12,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,35)(34,42)(36,44)(37,47)(38,43)(39,45)(40,46)(41,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,11,15,7)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,38,39,42)(34,35,43,45)(36,46,37,48)(40,47,41,44)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324420);