Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,19,15,17,14,29)(11,20,22,37,23,36,30,35)(12,24,26,34,28,39,27,18)(16,25,32,38,31,40,33,21);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,31)(18,35)(19,32)(20,24)(21,27)(26,38)(28,40)(29,33)(34,37)(36,39);(10,17)(16,31)(18,34)(19,29)(20,36)(24,39)(32,33)(35,37)Polycyclic presentation:
2^9|a^2=g,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,f^c=fh,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264484, #72264511, #72264525, … |
| maximal normal subgroup of | #72264484, #72264511, #72264525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401101);