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 | |
3731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,5,9)(6,10,14,12)(11,15,13,16)(17,18,22,26)(19,27,28,23)(20,24,31,29)(21,25,30,32)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3,7,5)(2,8,4,9)(6,11,14,13)(10,15,12,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,31)(27,28)(30,32)(34,40)(35,43)(41,42)(44,45);(1,3,7,5)(2,9,4,8)(6,11,14,13)(10,16,12,15)(17,19,22,28)(18,23,26,27)(20,30,31,21)(24,25,29,32)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,4,7,2)(3,8,5,9)(6,12,14,10)(11,15,13,16)(17,20,22,31)(18,24,26,29)(19,21,28,30)(23,32,27,25)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,5,7,3)(2,8,4,9)(6,13,14,11)(10,15,12,16)(17,18)(19,23)(20,24)(21,32)(22,26)(25,30)(27,28)(29,31)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,21)(18,25)(19,20)(22,30)(23,29)(24,27)(26,32)(28,31)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=h,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, 8646636);