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 | #45612 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,14)(4,8,13,16)(5,9,15,12)(17,18,23,28)(19,24,21,26)(20,30,29,25)(22,27,31,32)(33,34,39,44)(35,43,38,40)(36,46,45,41)(37,42,47,48);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,19,23,21)(18,24,28,26)(20,31,29,22)(25,27,30,32)(33,35,39,38)(34,40,44,43)(36,47,45,37)(41,48,46,42);(3,11)(4,13)(7,14)(8,16)(17,20,23,29)(18,25,28,30)(19,22,21,31)(24,32,26,27)(33,36)(34,41)(35,37)(38,47)(39,45)(40,42)(43,48)(44,46);(1,4)(2,8)(3,12)(5,7)(6,13)(9,11)(10,16)(14,15)(17,21,23,19)(18,26,28,24)(20,22,29,31)(25,32,30,27)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48);(1,5)(2,9)(3,8)(4,14)(6,15)(7,13)(10,12)(11,16)(17,22)(18,27)(19,29)(20,21)(23,31)(24,25)(26,30)(28,32)(33,37,39,47)(34,42,44,48)(35,45,38,36)(40,46,43,41);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,18)(19,24)(20,30)(21,26)(22,27)(23,28)(25,29)(31,32)(33,38,39,35)(34,43,44,40)(36,47,45,37)(41,48,46,42)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8812480);