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 | |
4842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,13)(14,16)(15,21)(18,23)(19,22)(20,24)(25,26)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(2,3)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,24)(22,23)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(10,16)(12,18)(15,17)(19,24)(26,28)(27,34)(33,36)(35,40);(1,4,5,8)(2,6,3,7)(9,12,14,18)(10,13,16,23)(11,19,21,24)(15,20,17,22)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(9,11)(10,17)(12,19)(13,22)(14,21)(15,16)(18,24)(20,23)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(9,13)(10,18)(11,20)(12,16)(14,23)(15,24)(17,19)(21,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817325);