Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483463 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,22,24,35)(13,26)(14,29)(15,19)(17,23,31,36)(18,27)(20,33,25,38)(21,32)(28,39)(30,34,40,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,19,29)(12,20,24,25)(13,21,28,27)(17,30,31,40)(18,26,32,39)(22,33,35,38)(23,34,36,37);(1,3,4,7)(2,6,5,8)(9,12,13,23,14,25,27,37,15,24,28,36,11,20,21,34)(10,17,18,22,16,30,26,33,19,31,32,35,29,40,39,38);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ef,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264744, #72264749, #72264848, … |
| maximal normal subgroup of | #72264744, #72264749, #72264848, … |
| maximal quotient of | #72343503, #72343568, #72343665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405103);