Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25076 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1692 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)(9,10,12,14)(11,15,16,13)(18,22)(19,24)(20,23);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(18,22)(19,24)(20,23);(10,14)(11,16)(17,18,23,24,21,19,20,22);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,19,20,24,21,18,23,22);(2,5)(6,8)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=ghi,c^2=hi,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72313233, #72313283, #72313513, … |
| maximal normal subgroup of | #72313233, #72313283, #72313513, … |
| maximal quotient of | #72319983, #72320298 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510057);