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 | |
3784 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)(17,18,23,27)(19,21,30,24)(20,31,29,25)(22,32,28,26)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,23,30)(18,24,27,21)(20,22,29,28)(25,32,31,26)(35,41)(38,45)(42,47)(46,48);(2,6)(7,10)(8,12)(14,16)(17,20,23,29)(18,25,27,31)(19,28,30,22)(21,32,24,26)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,32)(22,31)(23,24)(25,28)(26,29)(27,30);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,19)(18,21)(20,28)(22,29)(23,30)(24,27)(25,32)(26,31)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,28)(18,26,27,32)(19,29,30,20)(21,31,24,25)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=gi,f^2=h,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, 8663018);