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 | #2670 | C22 |
| 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) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,12)(7,9)(11,15)(13,16)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,11,16,14)(17,19,22,24)(18,23,26,20)(21,27,29,31)(25,30,32,28)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(2,9)(6,16)(8,10)(11,14)(21,29)(25,32)(27,31)(28,30)(34,39)(37,45)(40,41)(43,48);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,14,16,11)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,5,7,15)(2,11,9,14)(3,12,4,13)(6,10,16,8)(17,20)(18,24)(19,26)(21,28)(22,23)(25,31)(27,32)(29,30)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,10,15,8)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300873);