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 | #45740 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(7,10)(8,12)(14,16)(21,29)(25,30)(26,31)(28,32)(33,34,38,39)(35,43,45,40)(36,44,46,41)(37,47,48,42);(3,9)(7,10)(11,15)(14,16)(18,23)(20,27)(25,30)(28,32)(34,39)(36,46)(40,43)(42,47);(1,2,6,4)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,35,38,45)(34,40,39,43)(36,42,46,47)(37,44,48,41);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,19,22,24)(18,20,23,27)(21,26,29,31)(25,28,30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,45)(44,46)(47,48);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,36,38,46)(34,41,39,44)(35,42,45,47)(37,43,48,40);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,37)(34,42)(35,44)(36,43)(38,48)(39,47)(40,46)(41,45)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017102);