Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,17)(15,19)(18,20);(6,9)(10,12);(13,14,15,17,16,18,19,20);(2,4);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi152 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×#525, #72305409, #72305422, … |
| maximal normal subgroup of | SD16×#525, #72305409, #72305422, … |
| maximal quotient of | #72306153, #72321927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537458);