Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26)(27,36)(28,32)(29,35)(30,34)(31,39)(33,38)(37,40);(1,3,4,8)(2,5,6,7)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27,30,28)(26,32,34,36)(29,38,31,37)(33,35,40,39);(3,8)(5,7)(9,10)(11,15)(12,21)(13,16)(14,17)(18,20)(19,24)(22,23)(25,28,30,27)(26,33,34,40)(29,38,31,37)(32,35,36,39);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(26,34)(27,28)(35,39)(37,38);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,29,30,31)(26,35,34,39)(27,37,28,38)(32,33,36,40)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72283256, #72283313, #72283690, … |
| maximal normal subgroup of | #72283256, #72283313, #72283690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893687);