Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(9,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(5,6)(7,8)(14,18)(19,24)(20,25)(26,28);(2,4)(6,8)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(15,21)(19,24)(22,27)(26,28);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^9|a^2=i,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
| maximal subgroup of | #72304732, #72304737, #72310669, … |
| maximal normal subgroup of | #72304732, #72304737, #72310669, … |
| maximal quotient of | #72304995, #72304996, #72305063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 422225);