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 | |
4297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,9,11,6)(4,8,13,15)(10,16,12,14)(17,18,22,23)(19,24,27,21)(20,25,30,31)(26,32,28,29)(33,34,38,41)(35,37,44,47)(36,40,46,39)(42,48,43,45);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(18,23)(21,24)(25,31)(29,32)(33,35,38,44)(34,37,41,47)(36,43,46,42)(39,48,40,45);(2,7)(4,13)(6,9)(10,12)(17,19,22,27)(18,24,23,21)(20,28,30,26)(25,29,31,32)(33,36,38,46)(34,39,41,40)(35,42,44,43)(37,45,47,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,41)(42,44)(47,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,20,22,30)(18,25,23,31)(19,26,27,28)(21,29,24,32)(33,35)(34,37)(36,43)(38,44)(39,48)(40,45)(41,47)(42,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,13)(15,16)(17,21)(18,19)(20,29)(22,24)(23,27)(25,26)(28,31)(30,32)(33,37)(34,35)(36,45)(38,47)(39,42)(40,43)(41,44)(46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669272);