Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624184 |
| 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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(11,20)(12,14)(13,25)(16,29)(17,23)(19,33)(21,37)(22,36)(24,27)(26,28)(30,35)(31,40)(32,39)(34,38);(1,3)(2,5)(4,8)(6,7)(9,10,15,18)(11,16,22,31)(12,23,14,17)(13,19,27,34)(20,29,36,40)(21,32,28,30)(24,38,25,33)(26,35,37,39);(9,11)(10,16)(12,21)(13,26)(14,28)(15,22)(17,30)(18,31)(19,35)(20,25)(23,32)(24,36)(27,37)(29,33)(34,39)(38,40);(3,8)(5,7)(9,12,15,14)(10,17,18,23)(11,21,22,28)(13,24,27,25)(16,30,31,32)(19,33,34,38)(20,26,36,37)(29,39,40,35);(2,6)(5,7)(10,18)(16,31)(17,23)(19,34)(29,40)(30,32)(33,38)(35,39)Polycyclic presentation:
2^9|b^2=i,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72314433, #72315824, #72315871, … |
| maximal normal subgroup of | #72314433, #72315824, #72315871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531385);