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 | |
4149 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)(3,5)(4,9)(6,7)(8,10)(11,14)(12,15)(13,16)(33,34,39,41)(35,36,45,40)(37,42,47,46)(38,48,44,43);(2,7)(3,10)(4,12)(14,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35,39,45)(34,40,41,36)(37,44,47,38)(42,48,46,43);(3,10)(4,12)(5,8)(9,15)(18,22)(19,26)(20,29)(31,32)(34,41)(36,40)(37,47)(38,44);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,12)(15,16)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,36,39,40)(34,35,41,45)(37,48,47,43)(38,46,44,42);(1,4)(2,9)(3,11)(5,14)(6,12)(7,15)(8,16)(10,13)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,37)(34,42)(35,44)(36,43)(38,45)(39,47)(40,48)(41,46);(1,5,6,8)(2,3,7,10)(4,14,12,16)(9,11,15,13)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38,39,44)(34,43,41,48)(35,37,45,47)(36,46,40,42)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965694);