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×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,8,12,7)(4,13,14,5)(10,11,16,15)(17,18)(19,25)(20,24)(21,26)(22,27)(23,30)(28,29)(31,32)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(1,3,6,12)(2,7,9,8)(4,10,14,16)(5,11,13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(1,3,6,12)(2,7,9,8)(4,10,14,16)(5,11,13,15)(17,20,22,26)(18,24,27,21)(19,28,30,32)(23,31,25,29)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,3)(2,8)(4,10)(5,15)(6,12)(7,9)(11,13)(14,16)(17,20)(18,24)(19,28)(21,27)(22,26)(23,31)(25,29)(30,32)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,4,6,14)(2,5,9,13)(3,10,12,16)(7,11,8,15)(17,19)(18,25)(20,28)(21,31)(22,30)(23,27)(24,29)(26,32);(1,5)(2,4)(3,11)(6,13)(7,10)(8,16)(9,14)(12,15)(17,21,22,24)(18,26,27,20)(19,29,30,31)(23,32,25,28)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653207);