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 | |
3743 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,14)(15,16)(17,18)(19,25)(20,21)(22,24)(23,28)(26,27)(29,32)(30,31)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3,7,5)(2,8,12,9)(4,15,11,13)(6,14,10,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,25)(26,31)(30,32)(34,40)(35,43)(41,42)(44,45);(1,3,7,5)(2,9,12,8)(4,13,11,15)(6,14,10,16)(17,18,22,24)(19,23,28,25)(20,30,31,21)(26,27,29,32)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,4,7,11)(2,10,12,6)(3,13,5,15)(8,14,9,16)(17,20,22,31)(18,21,24,30)(19,26,28,29)(23,32,25,27)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,5,7,3)(2,8,12,9)(4,15,11,13)(6,16,10,14)(17,19)(18,23)(20,26)(21,32)(22,28)(24,25)(27,30)(29,31)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,6,7,10)(2,11,12,4)(3,14,5,16)(8,13,9,15)(17,21)(18,20)(19,27)(22,30)(23,29)(24,31)(25,26)(28,32)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=i,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, 8646588);