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 | #52416 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4089 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(18,23)(19,26)(25,30)(27,32);(1,3)(2,8)(4,13)(5,7)(6,10)(9,11)(12,14)(15,16)(17,18,22,23)(19,24,26,20)(21,25,29,30)(27,31,32,28)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(2,9)(3,5)(4,14)(6,16)(19,26)(20,24)(21,29)(25,30)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,15,16,10)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901734);