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 | |
3765 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(18,22)(23,28)(24,29)(30,32)(33,34,40,38)(35,43,47,41)(36,37,42,39)(44,45,48,46);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(20,27)(24,29)(25,31)(30,32)(33,35,40,47)(34,41,38,43)(36,48,42,44)(37,45,39,46);(1,2,5,8)(3,6,11,9)(4,7,13,12)(10,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,36)(34,39)(35,44)(37,38)(40,42)(41,45)(43,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,37,40,39)(34,36,38,42)(35,45,47,46)(41,48,43,44);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,38,40,34)(35,41,47,43)(36,37,42,39)(44,45,48,46);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,39,40,37)(34,42,38,36)(35,46,47,45)(41,44,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=ghi,e^2=h,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, 8662388);