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 | |
3990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31)(33,34,39,43)(35,44,46,40)(36,37,47,41)(38,48,45,42);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,15)(14,16)(17,18,22,26)(19,27,28,23)(20,21,25,24)(29,30,31,32)(33,35,39,46)(34,40,43,44)(36,45,47,38)(37,48,41,42);(2,6)(3,10)(4,13)(15,16)(19,28)(20,25)(21,24)(23,27)(33,36,39,47)(34,41,43,37)(35,45,46,38)(40,42,44,48);(1,3)(2,7)(4,14)(5,10)(6,9)(8,16)(11,13)(12,15)(17,19)(18,23)(20,31)(21,30)(22,28)(24,32)(25,29)(26,27)(33,37,39,41)(34,36,43,47)(35,42,46,48)(38,44,45,40);(1,4)(2,8)(3,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,20,22,25)(18,24,26,21)(19,29,28,31)(23,30,27,32)(33,36)(34,37)(35,45)(38,46)(39,47)(40,48)(41,43)(42,44);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,21,22,24)(18,25,26,20)(19,30,28,32)(23,29,27,31)(33,38,39,45)(34,42,43,48)(35,36,46,47)(37,44,41,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966551);