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 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,8,11,7)(5,12,13,9)(10,14,16,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,34,38,39)(35,36,40,44)(37,45,46,41)(42,48,47,43);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,35,38,40)(34,36,39,44)(37,42,46,47)(41,43,45,48);(2,4)(5,13)(7,8)(10,16)(20,28)(23,30)(26,32)(29,31)(34,39)(35,40)(37,46)(43,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,45)(46,48);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37)(34,41)(35,42)(36,43)(38,46)(39,45)(40,47)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9385529);