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 | |
4363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,25)(20,24)(21,26)(22,23)(27,30)(28,32)(29,31)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(4,11)(7,9)(12,16)(14,15)(18,23)(19,26)(20,28)(30,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,3)(7,9)(8,10)(14,15)(18,23)(19,26)(24,32)(27,29)(34,35)(39,41)(40,42)(46,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19)(18,21)(20,29)(22,26)(23,25)(24,30)(27,28)(31,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,24)(19,27)(21,31)(22,28)(23,32)(25,30)(26,29)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,19)(20,30)(22,25)(23,26)(24,29)(27,32)(28,31)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154307);