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 | |
4241 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)(19,25)(23,26)(27,31)(30,32)(33,34)(35,37)(36,41)(38,46)(39,40)(42,47)(43,48)(44,45);(2,7)(4,8)(5,14)(11,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,35,39,36)(34,37,40,41)(38,44,47,43)(42,48,46,45);(2,7)(3,9)(10,13)(11,16)(18,22)(23,26)(24,28)(30,32)(34,40)(35,36)(42,46)(43,44);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,39,35)(34,41,40,37)(38,44,47,43)(42,48,46,45);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,37,39,41)(34,36,40,35)(38,45,47,48)(42,44,46,43);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,38,39,47)(34,42,40,46)(35,43,36,44)(37,45,41,48)Polycyclic presentation:
2^9|b^2=gh,d^2=gh,e^2=h,f^2=i,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, 8783639);