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 |
| prime radical | #43478 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,8,14,15)(4,9,5,10)(6,11,16,13)(17,18,23,26)(19,24,29,22)(20,30,21,25)(27,32,28,31)(33,34,39,37)(35,36,43,40)(38,41,46,45)(42,44,48,47);(1,3,7,14)(2,8,12,15)(4,11,5,13)(6,9,16,10)(17,19,23,29)(18,24,26,22)(20,28,21,27)(25,32,30,31)(33,35,39,43)(34,40,37,36)(38,42,46,48)(41,47,45,44);(1,4,7,5)(2,9,12,10)(3,13,14,11)(6,15,16,8)(17,20,23,21)(18,25,26,30)(19,27,29,28)(22,32,24,31)(33,34,39,37)(35,40,43,36)(38,45,46,41)(42,44,48,47);(1,5,7,4)(2,10,12,9)(3,13,14,11)(6,15,16,8)(17,21,23,20)(18,25,26,30)(19,28,29,27)(22,32,24,31)(33,36)(34,35)(37,43)(38,44)(39,40)(41,42)(45,48)(46,47);(1,3)(2,8)(4,13)(5,11)(6,9)(7,14)(10,16)(12,15)(17,18)(19,22)(20,25)(21,30)(23,26)(24,29)(27,32)(28,31)(33,37,39,34)(35,40,43,36)(38,45,46,41)(42,47,48,44);(1,6,7,16)(2,11,12,13)(3,9,14,10)(4,8,5,15)(17,22)(18,19)(20,31)(21,32)(23,24)(25,28)(26,29)(27,30)(33,38,39,46)(34,41,37,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gi,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8290974);