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 | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,12,13,8)(10,14,16,15)(17,18,23,26)(19,22,29,24)(20,25,30,21)(27,28,31,32)(36,43)(40,45)(42,47)(46,48);(2,4)(7,9)(8,12)(14,15)(20,30)(21,25)(27,31)(28,32)(34,38)(35,41)(40,45)(42,47);(17,19,23,29)(18,24,26,22)(20,27,30,31)(21,28,25,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20)(18,25)(19,27)(21,26)(22,28)(23,30)(24,32)(29,31);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21,23,25)(18,20,26,30)(19,28,29,32)(22,31,24,27)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22)(18,19)(20,28)(21,31)(23,24)(25,27)(26,29)(30,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000518);