Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1790 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 850 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,12,15,13,16,21,24)(11,17,19,23,18,14,20,22);(1,2,4,6)(3,7,8,5)(10,14)(15,23)(16,17)(22,24);(1,3,4,8)(2,5,6,7)(11,18)(19,20)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=eh,g^b=gh,d^c=dhi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72282800, #72282829, #72282921, … |
| maximal normal subgroup of | #72282800, #72282829, #72282921, … |
| maximal quotient of | C2×#149480, #72303613, #72303614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56676);