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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3989 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)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(18,24)(19,22)(20,26)(29,31)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(1,3,5,11)(2,6,9,8)(4,10,13,16)(7,14,12,15)(17,18)(19,27)(20,21)(22,25)(23,24)(26,32)(28,29)(30,31)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,2,5,9)(3,8,11,6)(4,12,13,7)(10,14,16,15)(17,19,23,22)(18,25,24,27)(20,30,26,28)(21,29,32,31)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,20,23,26)(18,21,24,32)(19,28,22,30)(25,31,27,29);(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,13)(14,16)(17,21)(18,26)(19,29)(20,24)(22,31)(23,32)(25,28)(27,30)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,22,23,19)(18,25,24,27)(20,30,26,28)(21,31,32,29)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150748);