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 | |
4303 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)(3,5)(4,8)(6,7)(9,14)(10,13)(11,16)(12,15)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,29,28,31)(33,34,39,41)(35,44,45,40)(36,46,43,38)(37,42,47,48);(2,7)(3,9)(8,15)(10,16)(17,19,22,27)(18,21,25,23)(20,28,30,26)(24,29,32,31)(33,35)(34,40)(36,37)(38,42)(39,45)(41,44)(43,47)(46,48);(2,7)(3,9)(8,15)(10,16)(34,41)(35,45)(36,43)(42,48);(1,3,6,9)(2,5,7,14)(4,10,12,16)(8,13,15,11)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,30)(31,32)(33,36,39,43)(34,38,41,46)(35,37,45,47)(40,42,44,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,20)(18,24)(19,26)(21,31)(22,30)(23,29)(25,32)(27,28)(33,37)(34,42)(35,43)(36,45)(38,44)(39,47)(40,46)(41,48);(1,5)(2,9)(3,7)(4,11)(6,14)(8,10)(12,13)(15,16)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,28,32,26)(33,38)(34,43)(35,42)(36,41)(37,44)(39,46)(40,47)(45,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9845410);