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 | #53152 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3863 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)(3,12)(4,5)(6,8)(7,9)(10,11)(13,16)(14,15)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3,7,6)(2,8,9,12)(4,15,11,13)(5,16,10,14)(20,28)(23,30)(26,32)(29,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,9)(4,11)(8,12)(13,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4,7,11)(2,10,9,5)(3,13,6,15)(8,16,12,14)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,5,7,10)(2,11,9,4)(3,14,6,16)(8,15,12,13)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47);(1,6,7,3)(2,12,9,8)(4,15,11,13)(5,16,10,14)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022565);