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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3738 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)(4,12)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18,23,28)(19,24,30,21)(20,26,32,25)(22,31,29,27)(33,34,39,36)(35,37,40,42)(38,41,47,44)(43,45,48,46);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(17,19,23,30)(18,24,28,21)(20,29,32,22)(25,31,26,27)(35,40)(37,42)(43,48)(45,46);(17,20,23,32)(18,25,28,26)(19,22,30,29)(21,31,24,27)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,20)(18,26)(19,29)(21,31)(22,30)(23,32)(24,27)(25,28)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,5,6,10)(2,8,9,3)(4,13,14,16)(7,15,12,11)(17,21)(18,19)(20,31)(22,25)(23,24)(26,29)(27,32)(28,30)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,22,23,29)(18,27,28,31)(19,20,30,32)(21,26,24,25)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9770573);