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 | |
3749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,8,14,7)(4,5,10,9)(12,15,16,13)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,2)(3,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(18,23)(19,26)(25,28)(27,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,3,6,14)(2,7,11,8)(4,12,10,16)(5,13,9,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,3)(2,8)(4,12)(5,15)(6,14)(7,11)(9,13)(10,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4)(2,9)(3,12)(5,11)(6,10)(7,15)(8,13)(14,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48);(1,5,6,9)(2,10,11,4)(3,13,14,15)(7,16,8,12)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=i,e^2=gh,f^2=gi,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, 8658658);