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 | |
4447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39);(1,2,5,4)(3,7,8,6)(10,12)(15,17)(16,20)(23,24)(27,36)(28,38)(31,32)(33,39);(1,3,5,8)(2,6,4,7)(25,27)(26,31)(28,37)(29,33)(30,36)(32,35)(34,38)(39,40);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,22)(20,23)(25,27)(26,32)(28,37)(29,39)(30,36)(31,35)(33,40)(34,38);(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,28)(26,33)(27,37)(29,31)(30,38)(32,40)(34,36)(35,39);(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,29,30,40)(26,34,35,37)(27,33,36,39)(28,31,38,32)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416172);