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 | |
4113 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,25)(19,21,27,23)(20,24,30,32)(26,31,28,29)(34,40)(36,41)(42,47)(44,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,21)(20,28)(22,27)(23,25)(24,29)(26,30)(31,32)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47);(2,6)(7,13)(8,11)(14,16)(17,19,22,27)(18,23,25,21)(20,26,30,28)(24,29,32,31)(33,35,39,37)(34,36,40,41)(38,43,46,45)(42,44,47,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,31)(28,30)(29,32)(33,36,39,41)(34,35,40,37)(38,44,46,48)(42,43,47,45);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,30)(18,24,25,32)(19,26,27,28)(21,31,23,29)(33,37,39,35)(34,41,40,36)(38,43,46,45)(42,44,47,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21)(18,19)(20,29)(22,23)(24,28)(25,27)(26,32)(30,31)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619143);