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 | |
4163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,11,5)(4,7,12,15)(10,16,13,14)(17,18,22,20)(19,23,25,27)(21,24,30,28)(26,31,29,32)(33,34,39,42)(35,40,46,41)(36,47,37,38)(43,45,44,48);(1,3,6,11)(2,5,8,9)(4,10,12,13)(7,14,15,16)(17,19)(18,23)(20,27)(21,29)(22,25)(24,32)(26,30)(28,31)(33,35,39,46)(34,40,42,41)(36,44,37,43)(38,45,47,48);(4,12)(7,15)(10,13)(14,16)(19,25)(23,27)(26,29)(31,32)(33,36,39,37)(34,38,42,47)(35,43,46,44)(40,48,41,45);(1,3)(2,5)(4,13)(6,11)(7,16)(8,9)(10,12)(14,15)(17,20,22,18)(19,23,25,27)(21,28,30,24)(26,31,29,32)(33,37,39,36)(34,38,42,47)(35,44,46,43)(40,48,41,45);(1,4,6,12)(2,7,8,15)(3,10,11,13)(5,14,9,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,38)(34,36)(35,45)(37,42)(39,47)(40,44)(41,43)(46,48);(1,5)(2,3)(4,14)(6,9)(7,10)(8,11)(12,16)(13,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,46)(40,42)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259630);