Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1596 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1596 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(17,20);(5,7)(6,9,10,11)(8,12)(13,14)(15,17)(16,18)(19,20);(1,3,4,2)(13,15,16,19)(14,17,18,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,b^a=bd,c^a=ce,g^a=gh,d^b=dh,e^b=ei,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251455, #72251457, #72251461, … |
| maximal normal subgroup of | #72251455, #72251457, #72251461, … |
| maximal quotient of | #72255998, #72256126, #72256137 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17226);