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 | |
3746 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)(3,5)(4,13)(6,9)(7,10)(8,12)(11,14)(15,16)(17,18,23,28)(19,26,21,24)(20,30,31,25)(22,27,29,32)(33,34,39,36)(35,42,38,40)(37,41,45,44)(43,48,46,47);(1,3)(2,8)(4,11)(5,10)(6,16)(7,12)(9,14)(13,15)(17,19,23,21)(18,24,28,26)(20,22,31,29)(25,27,30,32)(37,45)(41,44)(43,46)(47,48);(1,4,7,6)(2,9,10,13)(3,11,12,16)(5,15,8,14)(17,20,23,31)(18,25,28,30)(19,29,21,22)(24,32,26,27)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,5,7,8)(2,3,10,12)(4,14,6,15)(9,16,13,11)(17,21,23,19)(18,26,28,24)(20,22,31,29)(25,27,30,32)(33,36,39,34)(35,42,38,40)(37,44,45,41)(43,48,46,47);(1,6,7,4)(2,9,10,13)(3,11,12,16)(5,14,8,15)(17,22)(18,27)(19,20)(21,31)(23,29)(24,25)(26,30)(28,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,3)(2,8)(4,11)(5,10)(6,16)(7,12)(9,14)(13,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,38,39,35)(34,42,36,40)(37,46,45,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113648);