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 | |
4044 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)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,28)(21,22)(23,27)(25,31)(26,32)(29,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,45)(44,48)(46,47);(3,9)(6,13)(10,15)(14,16)(17,19)(18,24)(20,21)(22,28)(23,29)(25,32)(26,31)(27,30)(34,39)(36,40)(42,47)(44,45);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,20,23,31)(18,25,27,28)(19,26,29,21)(22,30,32,24)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,45,47,44);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,25)(19,21)(22,30)(23,31)(24,32)(26,29)(27,28);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21)(18,22)(19,20)(23,26)(24,28)(25,30)(27,32)(29,31)(33,36,38,40)(34,41,39,35)(37,45,46,44)(42,43,47,48);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,22)(18,26)(19,28)(20,30)(21,27)(23,32)(24,31)(25,29)(33,37,38,46)(34,42,39,47)(35,43,41,48)(36,44,40,45)Polycyclic presentation:
2^9|c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817305);