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 | |
3774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,25)(20,28)(21,26)(22,23)(24,30)(27,29)(31,32)(33,34,39,43)(35,37,41,48)(36,42,38,40)(44,47,45,46);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(18,23)(19,26)(24,28)(27,32)(33,35)(34,37)(36,44)(38,45)(39,41)(40,46)(42,47)(43,48);(2,6)(7,10)(8,12)(14,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,31)(30,32)(33,36,39,38)(34,40,43,42)(35,44,41,45)(37,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,24,23,28)(19,27,26,32)(21,29,25,31);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,18)(19,21)(20,24)(22,23)(25,26)(27,29)(28,30)(31,32)(33,37)(34,41)(35,43)(36,46)(38,47)(39,48)(40,45)(42,44);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,32)(30,31)(33,38,39,36)(34,42,43,40)(35,45,41,44)(37,47,48,46)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=gh,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, 8663104);