Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483439 |
| 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 | C4×C8 | C24 |
| prime radical | C24 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(18,22,23,28)(19,25)(20,27)(24,31,30,29)(26,32);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,18,19,22,20,24,25,29,21,23,26,28,27,30,32,31);(3,9)(6,12)(10,15)(13,16);(2,7)(6,12)(8,14)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=g,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284874, #72287142, #72287171, … |
| maximal normal subgroup of | #72284874, #72287142, #72287171, … |
| maximal quotient of | #72343400, #72343518, #72343790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390635);