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 | |
4223 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,22,21)(19,25,26,23)(20,28,30,24)(27,31,29,32)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(2,7)(4,8)(5,14)(11,16)(18,21)(23,25)(24,28)(31,32)(33,35)(34,39)(36,37)(38,43)(40,46)(41,48)(42,44)(45,47);(2,7)(3,9)(10,13)(11,16)(19,26)(23,25)(27,29)(31,32)(35,43)(36,45)(39,44)(40,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,26)(18,23,21,25)(20,29,30,27)(24,32,28,31)(33,36,38,45)(34,40,42,48)(35,37,43,47)(39,46,44,41);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,22,30)(18,24,21,28)(19,27,26,29)(23,31,25,32)(33,37)(34,41)(35,36)(38,47)(39,48)(40,44)(42,46)(43,45);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,22,18)(19,25,26,23)(20,28,30,24)(27,32,29,31)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8745658);