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 | |
4274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,18)(12,16)(13,15)(14,17)(19,23)(20,22)(21,24)(25,26)(27,31)(28,32)(29,35)(30,33)(34,37)(36,39)(38,40);(2,3)(6,7)(9,11,14,13)(10,15,17,18)(12,20,21,19)(16,23,24,22)(25,27,30,35)(26,31,33,29)(28,34,38,36)(32,37,40,39);(1,3,5,2)(4,7,8,6)(25,27)(26,29)(28,34)(30,35)(31,33)(32,39)(36,38)(37,40);(1,4,5,8)(2,6,3,7)(9,12,14,21)(10,16,17,24)(11,19,13,20)(15,22,18,23)(25,27)(26,29)(28,36)(30,35)(31,33)(32,37)(34,38)(39,40);(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,28)(26,32)(27,34)(29,39)(30,38)(31,37)(33,40)(35,36);(9,13,14,11)(10,15,17,18)(12,20,21,19)(16,22,24,23)(25,29)(26,27)(28,37)(30,31)(32,36)(33,35)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=hi,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441564);