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 | #2673 | C22 |
| 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:
(1,2)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(18,23)(19,25)(24,30)(27,31)(33,34)(35,43)(36,47)(37,42)(38,39)(40,46)(41,45)(44,48);(1,3,7,5)(2,8,9,11)(4,12,14,13)(6,10,16,15)(17,18)(19,26)(20,25)(21,24)(22,23)(27,32)(28,31)(29,30)(33,35,39,46)(34,40,38,43)(36,37,45,44)(41,42,47,48);(2,9)(3,5)(4,14)(6,16)(19,25)(20,26)(21,29)(24,30)(34,38)(36,45)(37,44)(40,43);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,19,22,25)(18,20,23,26)(21,27,29,31)(24,28,30,32)(33,36)(34,41)(35,44)(37,46)(38,47)(39,45)(40,48)(42,43);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,15,16,10)(17,20)(18,19)(21,28)(22,26)(23,25)(24,27)(29,32)(30,31)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,47,43,41);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,21,22,29)(18,24,23,30)(19,27,25,31)(20,28,26,32)(33,38,39,34)(35,43,46,40)(36,47,45,41)(37,48,44,42)Polycyclic presentation:
2^9|b^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8966393);