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 | |
4031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,8,4,7)(5,9,13,16)(11,15,12,14)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34,39,44)(35,40,46,48)(36,37,42,41)(38,47,45,43);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35,39,46)(34,40,44,48)(36,45,42,38)(37,43,41,47);(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,36)(34,41)(35,45)(37,44)(38,46)(39,42)(40,47)(43,48);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,36)(34,37)(35,45)(38,46)(39,42)(40,43)(41,44)(47,48);(1,2)(3,8)(4,7)(5,9)(6,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37)(34,42)(35,43)(36,44)(38,40)(39,41)(45,48)(46,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38,39,45)(34,43,44,47)(35,36,46,42)(37,40,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10331776);