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 | |
3983 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)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,24,29,31)(20,26,30,25)(22,32,28,27)(34,35)(36,46)(38,47)(41,44);(3,9)(6,13)(10,15)(14,16)(17,19,23,29)(18,24,21,31)(20,22,30,28)(25,27,26,32)(33,34,39,35)(36,40,46,43)(37,44,48,41)(38,45,47,42);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,20,23,30)(18,25,21,26)(19,28,29,22)(24,32,31,27)(33,35,39,34)(36,40,46,43)(37,41,48,44)(38,45,47,42);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,26)(19,22)(21,25)(23,30)(24,32)(27,31)(28,29)(33,36)(34,40)(35,43)(37,47)(38,48)(39,46)(41,45)(42,44);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,21,23,18)(19,24,29,31)(20,25,30,26)(22,32,28,27)(33,37,39,48)(34,41,35,44)(36,47,46,38)(40,45,43,42);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,22)(18,27)(19,20)(21,32)(23,28)(24,25)(26,31)(29,30)(33,38,39,47)(34,42,35,45)(36,48,46,37)(40,44,43,41)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,e^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, 9150944);