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 | |
3929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,10,6,8)(4,12,14,9)(11,15,13,16)(20,28)(24,31)(26,32)(29,30)(34,40)(35,36)(37,47)(38,48);(1,3,7,6)(2,8,5,10)(4,11,14,13)(9,15,12,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,34,39,40)(35,41,36,46)(37,42,47,45)(38,43,48,44);(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,39,36)(34,41,40,46)(37,44,47,43)(38,45,48,42);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,39,35)(34,41,40,46)(37,44,47,43)(38,42,48,45);(1,5,7,2)(3,10,6,8)(4,9,14,12)(11,15,13,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37)(34,42)(35,44)(36,43)(38,41)(39,47)(40,45)(46,48);(1,6,7,3)(2,10,5,8)(4,13,14,11)(9,16,12,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,43)(35,45)(36,42)(37,41)(39,48)(40,44)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044520);