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 | |
4281 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,22)(21,26)(23,25)(24,30)(27,31)(29,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,30)(18,24,25,28)(20,29,27,26)(21,31,32,22)(33,35)(34,38)(36,45)(37,46)(39,42)(40,41)(43,44)(47,48);(3,10)(6,7)(11,16)(14,15)(18,25)(20,27)(24,28)(26,29)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,41,44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,27)(18,22,25,31)(19,29,30,26)(21,28,32,24)(33,36)(34,40)(35,45)(37,42)(38,41)(39,46)(43,47)(44,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,23,32)(18,26,25,29)(19,22,30,31)(20,24,27,28)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,40,44,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,22)(18,27)(19,21)(20,25)(23,31)(24,26)(28,29)(30,32)(33,38,39,44)(34,42,43,35)(36,41,46,48)(37,47,45,40)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9076531);