Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10)(11,16)(12,18)(13,26)(14,30)(15,20)(17,22)(19,28)(21,31)(23,34)(24,33)(25,37)(27,32)(29,40)(35,39)(36,38);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,24)(13,22)(14,21)(15,23)(17,26)(18,33)(19,32)(20,34)(25,39)(27,28)(29,38)(30,31)(35,37)(36,40);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,21)(13,27)(14,24)(15,23)(16,26)(18,32)(19,33)(20,36)(22,28)(25,38)(29,39)(30,35)(31,37)(34,40);(1,4,5,3)(2,7,6,8);(9,12,15,14)(10,18,20,30)(11,21,23,24)(13,25,29,28)(16,31,34,33)(17,32,36,35)(19,26,37,40)(22,27,38,39)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72216688, #72225000, #72260392, … |
| maximal normal subgroup of | #72216688, #72225000, #72260392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531164);