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 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,11)(10,12)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,28,32,30)(33,34,39,43)(35,41,37,40)(36,42,47,38)(44,45,46,48);(2,6)(7,14)(8,11)(15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35,39,37)(34,40,43,41)(36,44,47,46)(38,45,42,48);(1,2,5,6)(3,7,9,14)(4,8,13,11)(10,15,12,16)(17,19,22,29)(18,23,26,24)(20,27,25,32)(21,28,31,30)(33,35,39,37)(34,41,43,40)(36,46,47,44)(38,45,42,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19)(18,24)(20,27)(21,30)(22,29)(23,26)(25,32)(28,31)(33,36,39,47)(34,42,43,38)(35,44,37,46)(40,48,41,45);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20)(18,21)(19,27)(22,25)(23,28)(24,30)(26,31)(29,32)(33,37,39,35)(34,40,43,41)(36,46,47,44)(38,45,42,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21)(18,25)(19,28)(20,26)(22,31)(23,32)(24,27)(29,30)(33,38,39,42)(34,36,43,47)(35,45,37,48)(40,44,41,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8649096);