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 | |
4223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,8,10,7)(4,9,12,13)(11,15,16,14)(20,28)(24,29)(27,32)(30,31)(33,34)(35,43)(36,42)(37,47)(38,40)(39,41)(44,45)(46,48);(3,10)(4,12)(7,8)(9,13)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(33,35,40,39)(34,41,38,43)(36,44,47,46)(37,45,42,48);(1,3)(2,7)(4,11)(5,8)(6,10)(9,14)(12,16)(13,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,36,40,47)(34,42,38,37)(35,44,39,46)(41,48,43,45);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,16)(13,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,37)(34,36)(35,45)(38,47)(39,48)(40,42)(41,44)(43,46);(1,4,6,12)(2,9,5,13)(3,11,10,16)(7,14,8,15)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,38,40,34)(35,41,39,43)(36,37,47,42)(44,48,46,45);(1,5,6,2)(3,8,10,7)(4,13,12,9)(11,15,16,14)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)(33,39,40,35)(34,41,38,43)(36,46,47,44)(37,45,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=gh,f^2=gi,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, 8404103);