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 | |
4089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,23,27)(19,28,30,24)(20,22,26,32)(21,25,31,29)(33,34,38,39)(35,40,43,36)(37,45,47,41)(42,44,48,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,30)(18,24,27,28)(20,21,26,31)(22,29,32,25)(34,39)(36,40)(41,45)(44,46);(2,6)(7,9)(8,12)(14,15)(33,35,38,43)(34,36,39,40)(37,42,47,48)(41,44,45,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,26)(18,22,27,32)(19,21,30,31)(24,29,28,25)(33,35)(34,40)(36,39)(37,42)(38,43)(41,46)(44,45)(47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21)(18,25)(19,26)(20,30)(22,24)(23,31)(27,29)(28,32)(33,36,38,40)(34,35,39,43)(37,46,47,44)(41,48,45,42);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22)(18,26)(19,29)(20,27)(21,28)(23,32)(24,31)(25,30)(33,37,38,47)(34,41,39,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,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, 9633754);