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 | |
4109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34,38,37)(35,39,43,41)(36,44,46,40)(42,48,45,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,34,38,37)(35,41,43,39)(36,40,46,44)(42,48,45,47);(4,12)(7,14)(10,16)(13,15)(18,22)(24,26)(25,28)(31,32)(34,37)(39,41)(40,44)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,21,23,19)(18,26,22,24)(20,27,30,29)(25,31,28,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,37,38,34)(35,41,43,39)(36,44,46,40)(42,48,45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8230689);