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 | |
3773 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)(3,8)(4,6)(5,14)(7,9)(10,13)(11,16)(12,15)(18,22)(20,27)(23,28)(26,31)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3)(2,8)(4,13)(5,11)(6,10)(7,12)(9,15)(14,16)(19,25)(23,28)(26,31)(30,32)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(2,9)(3,12)(4,11)(10,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,10,15,14)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5,7,13)(2,10,9,14)(3,11,12,4)(6,15,16,8)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6)(2,11)(3,14)(4,9)(5,15)(7,16)(8,13)(10,12)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662245);