Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,22)(21,24);(1,2,3,4)(10,14)(13,20)(15,21)(17,23);(9,11)(10,15)(12,18)(13,19)(14,21)(16,17)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=di,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=eh,h^c=hi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284336, #72284367, #72284422, … |
| maximal normal subgroup of | #72284336, #72284367, #72284422, … |
| maximal quotient of | #72284646, #72284737, #72286148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55363);