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 | |
3871 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)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,23,24)(19,25,20,21)(22,26,31,32)(27,30,28,29);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(18,24)(19,20)(26,32)(27,28)(34,38)(39,45)(40,46)(47,48);(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,36,37,44)(34,40,38,46)(35,41,42,43)(39,47,45,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003616);