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 | #39212 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,8,14,12)(10,16,13,15)(17,18,23,22)(19,24,27,28)(20,25,29,31)(21,26,30,32)(33,34,39,38)(35,43,37,40)(36,41,46,42)(44,48,45,47);(2,6)(7,9)(8,12)(15,16)(17,19)(18,24)(20,21)(22,28)(23,27)(25,26)(29,30)(31,32)(33,35,39,37)(34,40,38,43)(36,44,46,45)(41,47,42,48);(2,6)(7,9)(8,12)(15,16)(19,27)(20,29)(24,28)(25,31)(33,36,39,46)(34,41,38,42)(35,44,37,45)(40,47,43,48);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,30)(28,32)(33,36)(34,42)(35,44)(37,45)(38,41)(39,46)(40,48)(43,47);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21)(18,26)(19,20)(22,32)(23,30)(24,25)(27,29)(28,31)(33,37,39,35)(34,40,38,43)(36,45,46,44)(41,47,42,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22,23,18)(19,28,27,24)(20,25,29,31)(21,26,30,32)(33,38,39,34)(35,43,37,40)(36,42,46,41)(44,48,45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8629462);