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 | |
4092 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32)(35,41)(39,42)(43,47)(46,48);(3,9)(7,13)(10,15)(14,16)(18,23)(19,26)(25,31)(27,32)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(2,6)(7,13)(8,11)(14,16)(20,28)(25,31)(27,32)(29,30)(34,36)(39,42)(40,44)(46,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,29,31,30)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,25)(19,27)(21,30)(22,28)(23,31)(24,29)(26,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,24)(18,26,23,19)(20,29,28,30)(25,27,31,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9976418);