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 | |
3918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,39,37)(35,43,46,40)(36,47,45,41)(38,48,44,42);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,42)(47,48);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,34,39,37)(35,40,46,43)(36,47,45,41)(38,42,44,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,42,43,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,37,39,34)(35,43,46,40)(36,41,45,47)(38,48,44,42);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,41,43,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064158);