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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3955 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)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(17,18,23,22)(19,24,27,28)(20,25,29,31)(21,26,30,32)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,38,44,43);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,14)(13,16)(17,19)(18,24)(20,21)(22,28)(23,27)(25,26)(29,30)(31,32)(33,35,39,47)(34,40,45,48)(36,38,46,43)(37,42,44,41);(4,13)(7,12)(11,16)(14,15)(19,27)(20,29)(24,28)(25,31)(33,36,39,46)(34,41,45,42)(35,43,47,38)(37,40,44,48);(1,4,6,13)(2,7,10,12)(3,11,9,16)(5,15,8,14)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,30)(28,32)(33,36)(34,42)(35,43)(37,40)(38,47)(39,46)(41,45)(44,48);(1,3)(2,8)(4,11)(5,10)(6,9)(7,14)(12,15)(13,16)(17,21)(18,26)(19,20)(22,32)(23,30)(24,25)(27,29)(28,31)(33,37)(34,43)(35,42)(36,40)(38,45)(39,44)(41,47)(46,48);(1,5,6,8)(2,9,10,3)(4,14,13,15)(7,11,12,16)(17,22,23,18)(19,28,27,24)(20,25,29,31)(21,26,30,32)(33,38,39,43)(34,44,45,37)(35,46,47,36)(40,41,48,42)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^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, 8629448);