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 | |
1868 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1868 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) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,11,12,10)(8,9)(13,14)(15,21,23,18)(16,19)(17,20)(22,25,28,26)(24,27);(5,7,9,12)(6,10,8,11)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,28)(25,27);(2,4)(5,8,9,6)(7,11,12,10)(13,16,17,24)(14,19,20,27)(15,22,23,28)(18,25,21,26)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,g^b=gh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72268301, #72268310, #72268314, … |
| maximal normal subgroup of | #72268301, #72268310, #72268314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18656);