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 | |
3990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(8,15)(10,13)(11,16)(19,27)(20,29)(22,28)(24,32)(34,36)(35,43)(38,48)(41,46);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(17,18,23,26)(19,28,27,22)(20,24,29,32)(21,25,31,30)(33,34,39,36)(35,44,43,40)(37,46,45,41)(38,42,48,47);(2,7)(5,12)(8,15)(11,16)(17,19,23,27)(18,22,26,28)(20,21,29,31)(24,30,32,25)(33,35,39,43)(34,40,36,44)(37,38,45,48)(41,42,46,47);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(17,20)(18,24)(19,21)(22,30)(23,29)(25,28)(26,32)(27,31)(33,36,39,34)(35,44,43,40)(37,46,45,41)(38,47,48,42);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,21)(18,25)(19,29)(20,27)(22,24)(23,31)(26,30)(28,32)(33,37,39,45)(34,41,36,46)(35,38,43,48)(40,42,44,47);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,22)(18,19)(20,30)(21,24)(23,28)(25,29)(26,27)(31,32)(33,38)(34,42)(35,45)(36,47)(37,43)(39,48)(40,46)(41,44)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833426);