Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#468 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2062 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)(13,14)(15,24)(16,21)(17,22)(18,23)(19,25)(20,26)(27,28);(5,6,8,9)(7,10,11,12)(13,15,17,26,18,27,16,25)(14,19,21,28,23,20,22,24);(2,4)(6,9)(7,11)(13,15,16,25,18,27,17,26)(14,20,22,28,23,19,21,24);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,21,23,22)(15,25,27,26)(19,28,20,24);(6,9)(7,11)Polycyclic presentation:
2^9|b^2=fh,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ehi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257264, #72257285, #72263879, … |
| maximal normal subgroup of | #72257264, #72257285, #72263879, … |
| maximal quotient of | #72264917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6505478);