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 | |
3742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,11,12,7)(4,8,15,13)(5,9,14,16)(17,18)(19,25)(20,31)(21,27)(22,32)(23,28)(24,29)(26,30)(33,34)(35,36)(37,42)(38,43)(39,40)(41,48)(44,46)(45,47);(1,2)(3,7)(4,13)(5,9)(6,10)(8,15)(11,12)(14,16)(17,19,24,22)(18,25,29,32)(20,23,30,27)(21,31,28,26)(34,40)(35,41)(42,43)(44,45);(1,3,6,12)(2,7,10,11)(4,14,15,5)(8,9,13,16)(17,20)(18,26)(19,23)(21,25)(22,27)(24,30)(28,32)(29,31)(33,35,39,41)(34,36,40,48)(37,44,38,45)(42,46,43,47);(1,4,6,15)(2,8,10,13)(3,5,12,14)(7,16,11,9)(17,21,24,28)(18,27,29,23)(19,26,22,31)(20,25,30,32)(33,36)(34,41)(35,40)(37,46)(38,47)(39,48)(42,45)(43,44);(1,2)(3,7)(4,8)(5,16)(6,10)(9,14)(11,12)(13,15)(17,22,24,19)(18,25,29,32)(20,27,30,23)(21,31,28,26)(33,37,39,38)(34,42,40,43)(35,44,41,45)(36,46,48,47);(1,5)(2,9)(3,4)(6,14)(7,13)(8,11)(10,16)(12,15)(17,23,24,27)(18,28,29,21)(19,30,22,20)(25,26,32,31)(33,38,39,37)(34,43,40,42)(35,45,41,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646399);