Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 | |
1405 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)(11,19)(12,23)(13,28)(14,16)(15,20)(17,26)(18,25)(21,39)(22,36)(24,34)(27,35)(29,33)(30,38)(31,40)(32,37);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,31)(22,30)(23,26)(24,33)(25,28)(27,32)(29,34)(35,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,11,14,22)(10,15,16,30)(12,24,27,40)(13,29,25,21)(17,33,32,39)(18,34,28,31)(19,35,36,23)(20,37,38,26);(1,4,5,3)(2,7,6,8);(10,16)(15,30)(17,32)(18,28)(19,36)(23,35)(31,34)(33,39)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72322566 |
| maximal normal subgroup of | #72322566 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531176);