Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470077 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,23)(22,24);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16);(11,15)(13,16)(18,21)(19,24)(22,23);(11,15)(13,16);(1,3,4,7)(2,6,5,8)(10,14)(13,16)Polycyclic presentation:
2^9|e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305425, #72305459, #72305463 |
| maximal normal subgroup of | #72305425, #72305459, #72305463 |
| maximal quotient of | #72305618, #72305751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378944);