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 | |
3809 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)(3,7)(4,10)(5,15)(6,8)(9,11)(12,13)(14,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,39,36)(35,43,45,40)(37,41,44,46)(38,42,48,47);(1,3,6,11)(2,7,8,9)(4,14,15,12)(5,13,10,16)(17,19)(18,23)(20,22)(21,27)(24,31)(25,26)(28,30)(29,32)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,47,46,42);(2,8)(3,11)(4,15)(13,16)(17,18,22,25)(19,23,20,26)(21,24,30,29)(27,31,28,32)(33,34,39,36)(35,40,45,43)(37,46,44,41)(38,42,48,47);(1,3)(2,9)(4,12)(5,13)(6,11)(7,8)(10,16)(14,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,36,39,34)(35,40,45,43)(37,46,44,41)(38,47,48,42);(1,4,6,15)(2,10,8,5)(3,12,11,14)(7,13,9,16)(17,20,22,19)(18,23,25,26)(21,27,30,28)(24,32,29,31)(33,37)(34,41)(35,38)(36,46)(39,44)(40,47)(42,43)(45,48);(1,5,6,10)(2,4,8,15)(3,13,11,16)(7,14,9,12)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,41)(43,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9948059);