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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,16)(4,14,12,6)(5,9,15,13)(17,18)(19,23)(20,26)(21,29)(22,25)(24,30)(27,32)(28,31)(33,34,40,38)(35,43,47,41)(36,37,42,39)(44,45,48,46);(1,3)(2,8)(4,5)(6,13)(7,11)(9,14)(10,16)(12,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35,40,47)(34,41,38,43)(36,48,42,44)(37,45,39,46);(3,11)(4,12)(6,14)(8,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36)(34,39)(35,44)(37,38)(40,42)(41,45)(43,46)(47,48);(1,4)(2,6)(3,5)(7,12)(8,13)(9,16)(10,14)(11,15)(33,37,40,39)(34,36,38,42)(35,45,47,46)(41,48,43,44);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,13)(14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,38,40,34)(35,41,47,43)(36,37,42,39)(44,45,48,46);(1,6)(2,4)(3,13)(5,8)(7,14)(9,11)(10,12)(15,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,39,40,37)(34,42,38,36)(35,46,47,45)(41,44,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=g,e^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=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835947);