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 | #40244 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,15)(4,9,13,14)(5,6,10,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,31)(28,30)(29,32)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,3,7,12)(2,8,11,15)(4,5,13,10)(6,14,16,9)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(3,12)(4,13)(8,15)(9,14)(17,18,21,25)(19,23,27,22)(20,24,30,28)(26,32,29,31)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,4,7,13)(2,9,11,14)(3,5,12,10)(6,15,16,8)(17,18)(19,23)(20,28)(21,25)(22,27)(24,30)(26,31)(29,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,5)(2,6)(3,13)(4,12)(7,10)(8,14)(9,15)(11,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,31)(26,30)(28,32)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,6)(2,10)(3,14)(4,8)(5,11)(7,16)(9,12)(13,15)(17,20,21,30)(18,24,25,28)(19,26,27,29)(22,31,23,32)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8625505);