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 | #2688 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3880 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(18,19)(23,25)(24,26)(31,32)(33,34,39,42)(35,37,41,47)(36,38,46,40)(43,48,44,45);(2,7)(4,10)(9,13)(12,14)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32)(33,35,39,41)(34,37,42,47)(36,44,46,43)(38,45,40,48);(2,7)(3,8)(9,13)(11,16)(33,36,39,46)(34,40,42,38)(35,43,41,44)(37,45,47,48);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,36)(34,38)(35,44)(37,45)(39,46)(40,42)(41,43)(47,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37,39,47)(34,41,42,35)(36,45,46,48)(38,43,40,44);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,38)(34,36)(35,45)(37,44)(39,40)(41,48)(42,46)(43,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592733);