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 | |
3833 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)(4,12,13,9)(11,15,14,16)(18,20)(19,25)(21,29)(30,32)(33,34,39,35)(36,43,46,40)(37,41,38,42)(44,48,45,47);(1,3,7,6)(2,8,5,10)(4,11,13,14)(9,15,12,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(34,35)(40,43)(41,42)(47,48);(4,13)(9,12)(11,14)(15,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35,39,34)(36,43,46,40)(37,42,38,41)(44,48,45,47);(1,4,7,13)(2,9,5,12)(3,11,6,14)(8,15,10,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36,39,46)(34,40,35,43)(37,44,38,45)(41,47,42,48);(1,5,7,2)(3,10,6,8)(4,9,13,12)(11,15,14,16)(17,20,22,18)(19,23,25,26)(21,28,29,24)(27,30,31,32)(33,37,39,38)(34,41,35,42)(36,44,46,45)(40,47,43,48);(1,6,7,3)(2,10,5,8)(4,14,13,11)(9,16,12,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,37)(34,42,35,41)(36,45,46,44)(40,48,43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,d^2=g,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,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, 8066172);