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 | |
3958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,18,24,22)(19,25,30,29)(20,31,21,26)(23,32,28,27)(34,39)(36,43)(41,45)(44,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,24,30)(18,25,22,29)(20,28,21,23)(26,32,31,27)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,46)(47,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,20,24,21)(18,26,22,31)(19,28,30,23)(25,32,29,27)(33,35,38,40)(34,36,39,43)(37,42,46,47)(41,44,45,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,21,24,20)(18,26,22,31)(19,23,30,28)(25,32,29,27)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,48)(44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,22,24,18)(19,29,30,25)(20,31,21,26)(23,32,28,27)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,45)(46,48);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,23)(18,27)(19,20)(21,30)(22,32)(24,28)(25,26)(29,31)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358795);