Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10336 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1828 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1828 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(1,2,3,4)(5,6,9,7)(8,10,12,11)(17,20);(5,7,9,6)(8,10)(11,12)(13,14,16,18)(15,17)(19,20);(5,8)(6,10)(7,11)(9,12)(13,15)(14,17,18,20)(16,19)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=e,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264336, #72264494, C4×#10337 |
| maximal normal subgroup of | #72264336, #72264494, C4×#10337 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28605);