Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129431 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
890 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 142 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)(11,15,12,17)(13,18)(14,19)(16,20,23,21)(22,24);(1,2,4,6)(3,7,8,5)(10,15,19,17)(13,22)(16,24,23,18)(20,21);(1,3,4,8)(2,5,6,7)(9,11,14,12)(10,16)(13,20,22,21)(15,18)(17,24)(19,23)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fi,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=ghi,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72234078, #72234081, #72260917, … |
| maximal normal subgroup of | #72234078, #72234081, #72260917, … |
| maximal quotient of | #72234073, #72260543, #72261429 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56635);