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 | |
4021 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18,23,26)(19,24,20,21)(22,25,31,32)(27,30,28,29)(33,34)(35,41)(36,47)(37,48)(38,42)(39,40)(43,44)(45,46);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19,23,20)(18,21,26,24)(22,27,31,28)(25,29,32,30)(33,35)(34,41)(36,37)(38,44)(39,45)(40,46)(42,43)(47,48);(2,8)(4,9)(7,14)(10,15)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,30,32,29)(33,36,40,38)(34,42,39,47)(35,44,46,37)(41,48,45,43);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20,23,19)(18,21,26,24)(22,28,31,27)(25,29,32,30)(33,37,40,44)(34,43,39,48)(35,36,46,38)(41,42,45,47);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21)(18,19)(20,26)(22,30)(23,24)(25,28)(27,32)(29,31)(33,38,40,36)(34,42,39,47)(35,44,46,37)(41,43,45,48);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)(33,39,40,34)(35,45,46,41)(36,47,38,42)(37,48,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9837449);