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 | |
4135 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,47)(42,46)(44,48);(3,9)(7,13)(10,15)(14,16)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(34,39)(36,40)(41,45)(44,46);(2,6)(7,13)(8,11)(14,16)(18,25)(19,28)(20,30)(26,32)(33,35,38,43)(34,40,39,36)(37,42,47,48)(41,46,45,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,36)(34,35)(37,46)(38,40)(39,43)(41,48)(42,45)(44,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,37,38,47)(34,41,39,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|b^2=i,c^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9634250);