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 |
| prime radical | #43964 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4289 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)(3,7)(4,13)(5,8)(6,9)(10,16)(11,14)(12,15)(17,18,23,22)(19,27,29,24)(20,26,21,25)(28,31,30,32)(33,34,39,43)(35,44,46,40)(36,37,45,41)(38,42,48,47);(1,3,6,12)(2,7,9,15)(4,11,5,10)(8,16,13,14)(17,19)(18,24)(20,28)(21,30)(22,27)(23,29)(25,31)(26,32)(33,35,39,46)(34,40,43,44)(36,38,45,48)(37,47,41,42);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(17,20,23,21)(18,25,22,26)(19,28,29,30)(24,31,27,32)(33,34,39,43)(35,40,46,44)(36,41,45,37)(38,42,48,47);(1,5,6,4)(2,8,9,13)(3,11,12,10)(7,14,15,16)(17,21,23,20)(18,26,22,25)(19,28,29,30)(24,31,27,32)(33,36)(34,41)(35,38)(37,43)(39,45)(40,42)(44,47)(46,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,22,23,18)(19,24,29,27)(20,26,21,25)(28,31,30,32)(33,37,39,41)(34,36,43,45)(35,42,46,47)(38,40,48,44);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,16)(12,15)(17,19)(18,27)(20,28)(21,30)(22,24)(23,29)(25,32)(26,31)(33,38)(34,42)(35,45)(36,46)(37,40)(39,48)(41,44)(43,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8386690);