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 | |
4204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,14)(4,8,12,15)(10,13,16,11)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(2,7)(3,9)(8,15)(10,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(35,41)(38,45)(42,47)(46,48);(2,7)(3,9)(8,15)(10,16)(18,25)(19,20)(26,32)(27,28)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,5)(2,3)(4,11)(6,14)(7,9)(8,16)(10,15)(12,13)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=i,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, 9854224);