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 | |
4001 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,34,38,39)(35,41,43,36)(37,45,47,40)(42,44,46,48);(2,7)(5,8)(9,14)(13,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,35,38,43)(34,36,39,41)(37,46,47,42)(40,48,45,44);(4,12)(9,14)(10,16)(13,15)(18,22)(19,26)(24,28)(27,32)(34,39)(36,41)(40,45)(44,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,34)(35,36)(37,40)(38,39)(41,43)(42,44)(45,47)(46,48);(1,4,6,12)(2,9,7,14)(3,10,11,16)(5,13,8,15)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,46)(47,48);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,47)(34,40,39,45)(35,42,43,46)(36,44,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,e^2=gh,f^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=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150914);