Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100203 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| 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 | |
1540 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1540 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14)(15,18)(16,19)(17,20)(21,26)(22,25)(23,27)(24,28);(1,2,3,4)(13,15)(14,18)(16,21)(17,22)(19,26)(20,25)(23,28)(24,27);(5,7)(6,9)(8,12)(10,11)(13,16,17,23)(14,19,20,27)(15,21,22,28)(18,24,25,26)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72251433, #72251439, #72251450, … |
| maximal normal subgroup of | #72251433, #72251439, #72251450, … |
| maximal quotient of | #72255787, #72256039 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22920);