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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,6)(3,12,14,9)(4,10,5,15)(7,16,13,11)(18,21)(19,25)(20,28)(30,32)(33,34,39,43)(35,36,46,40)(37,41,38,42)(44,47,45,48);(1,3)(2,9)(4,7)(5,13)(6,12)(8,14)(10,11)(15,16)(17,18,22,21)(19,23,25,27)(20,29,28,24)(26,32,31,30)(33,35,39,46)(34,40,43,36)(37,44,38,45)(41,48,42,47);(1,4,8,5)(2,10,6,15)(3,7,14,13)(9,11,12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35)(34,36)(37,44)(38,45)(39,46)(40,43)(41,47)(42,48);(1,5,8,4)(2,10,6,15)(3,13,14,7)(9,11,12,16)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,36)(34,35)(37,47)(38,48)(39,40)(41,44)(42,45)(43,46);(1,6,8,2)(3,12,14,9)(4,15,5,10)(7,16,13,11)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,37,39,38)(34,41,43,42)(35,44,46,45)(36,47,40,48);(1,7,8,13)(2,11,6,16)(3,4,14,5)(9,10,12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,38,39,37)(34,42,43,41)(35,45,46,44)(36,48,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=gi,e^2=g,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, 8050350);