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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,5,11,7)(4,8,12,15)(10,13,16,14)(17,18,22,24)(19,20,26,28)(21,23,30,29)(25,27,32,31)(33,34,38,39)(35,36,40,43)(37,41,46,47)(42,44,48,45);(4,12)(8,15)(10,16)(13,14)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,31)(30,32)(34,39)(36,43)(41,47)(44,45);(1,3,6,11)(2,7,9,5)(4,10,12,16)(8,14,15,13)(17,20,22,28)(18,19,24,26)(21,27,30,31)(23,25,29,32)(33,35,38,40)(34,36,39,43)(37,42,46,48)(41,44,47,45);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,16)(14,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(1,4,6,12)(2,8,9,15)(3,10,11,16)(5,13,7,14)(17,21,22,30)(18,23,24,29)(19,25,26,32)(20,27,28,31)(33,36)(34,40)(35,39)(37,45)(38,43)(41,42)(44,46)(47,48);(1,5)(2,3)(4,13)(6,7)(8,10)(9,11)(12,14)(15,16)(17,18)(19,20)(21,29)(22,24)(23,30)(25,31)(26,28)(27,32)(33,37,38,46)(34,41,39,47)(35,42,40,48)(36,44,43,45)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818157);