Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C26×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C2 | C22×C43 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(5,6,7,8)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(9,10,11,12)(15,22)(18,21)(23,28)(25,27);(14,19)(18,21)(20,26)(25,27);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=f,b^a=bf,d^a=df,c^b=cf272 conjugacy classes — view the full character table.
| maximal subgroup of | #72239188, #72239189, #72239190, … |
| maximal normal subgroup of | #72239188, #72239189, #72239190, … |
| maximal quotient of | #72239227 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420525);