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 | #40673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,26)(20,21)(22,24)(23,27)(25,31)(28,32)(29,30)(33,34,39,44)(35,40,46,47)(36,43,38,41)(37,42,48,45);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,23,22)(18,24,27,26)(20,28,31,30)(21,29,25,32)(33,35,39,46)(34,40,44,47)(36,42,38,45)(37,43,48,41);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,20,23,31)(18,21,27,25)(19,28,22,30)(24,29,26,32)(33,36,39,38)(34,41,44,43)(35,45,46,42)(37,40,48,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20)(18,25)(19,28)(21,27)(22,30)(23,31)(24,32)(26,29)(33,37,39,48)(34,42,44,45)(35,41,46,43)(36,47,38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,21,23,25)(18,20,27,31)(19,29,22,32)(24,28,26,30)(33,38,39,36)(34,43,44,41)(35,45,46,42)(37,40,48,47);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,22,23,19)(18,26,27,24)(20,30,31,28)(21,32,25,29)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gi,d^2=i,e^2=gh,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=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014803);