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 | |
3908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,12,13,8)(4,9,14,16)(5,6,10,15)(17,18)(19,28)(20,25)(21,29)(22,30)(23,26)(24,27)(31,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(3,13)(4,14)(8,12)(9,16)(17,19,24,26)(18,23,27,28)(20,29,31,22)(21,25,30,32)(34,36)(39,41)(40,44)(46,47);(1,3)(2,8)(4,10)(5,14)(6,9)(7,13)(11,12)(15,16)(17,20,24,31)(18,25,27,32)(19,29,26,22)(21,23,30,28)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4,7,14)(2,9,11,16)(3,10,13,5)(6,12,15,8)(17,21)(18,22)(19,25)(20,23)(24,30)(26,32)(27,29)(28,31)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,5,7,10)(2,6,11,15)(3,14,13,4)(8,9,12,16)(17,22)(18,21)(19,20)(23,25)(24,29)(26,31)(27,30)(28,32);(1,6)(2,10)(3,9)(4,8)(5,11)(7,15)(12,14)(13,16)(17,23)(18,26)(19,27)(20,30)(21,31)(22,25)(24,28)(29,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8353035);