Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9383 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1306 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1306 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(14,17,18,19);(6,9,11,10)(13,14)(15,19)(16,18)(17,20);(1,2,3,4)(15,20)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di98 conjugacy classes — view the full character table.
| maximal subgroup of | #72227435, #72228197, C2×#110560, … |
| maximal normal subgroup of | #72227435, #72228197, C2×#110560, … |
| maximal quotient of | #72227431, C2×#110560, #72268608 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17756);