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 | #42359 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4049 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)(3,12)(4,5)(6,13)(7,11)(8,15)(9,16)(10,14)(17,18,22,26)(19,27,29,23)(20,30,31,24)(21,28,32,25)(33,34,39,44)(35,40,46,47)(36,43,38,41)(37,42,48,45);(1,3)(2,8)(4,13)(5,14)(6,9)(7,15)(10,16)(11,12)(17,19)(18,23)(20,25)(21,24)(22,29)(26,27)(28,31)(30,32)(33,35,39,46)(34,40,44,47)(36,42,38,45)(37,43,48,41);(1,4,7,16)(2,5,11,9)(3,13,15,10)(6,8,14,12)(17,20,22,31)(18,24,26,30)(19,25,29,28)(21,27,32,23)(33,36,39,38)(34,41,44,43)(35,45,46,42)(37,40,48,47);(1,4)(2,9)(3,13)(5,11)(6,8)(7,16)(10,15)(12,14)(17,20)(18,24)(19,28)(21,27)(22,31)(23,32)(25,29)(26,30)(33,37,39,48)(34,42,44,45)(35,41,46,43)(36,47,38,40);(1,5,7,9)(2,4,11,16)(3,14,15,6)(8,13,12,10)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,32)(30,31)(33,38,39,36)(34,43,44,41)(35,45,46,42)(37,40,48,47);(1,6,7,14)(2,10,11,13)(3,9,15,5)(4,8,16,12)(17,21,22,32)(18,25,26,28)(19,24,29,30)(20,27,31,23)(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=i,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, 8014802);