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 | |
3817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,11)(4,6,15,10)(12,16,14,13)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,34,39,36)(35,43,45,40)(37,41,44,46)(38,42,48,47);(1,3)(2,8)(4,14)(5,7)(6,13)(9,11)(10,16)(12,15)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,47,46,42);(2,9)(3,5)(4,15)(13,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,34,39,36)(35,40,45,43)(37,46,44,41)(38,42,48,47);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,13)(11,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36,39,34)(35,40,45,43)(37,46,44,41)(38,47,48,42);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,13,10,16)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,37)(34,41)(35,38)(36,46)(39,44)(40,47)(42,43)(45,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,41)(43,46)Polycyclic presentation:
2^9|a^2=gi,b^2=g,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, 9948041);