Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3716 conjugacy classes of subgroups; only the 388 normal subgroups are shown.
This group has 3716 conjugacy classes of subgroups — too many to draw. Only its 388 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,14,12,13,15,16);(1,3)(2,5)(4,8)(6,7);(1,2)(3,5)(4,6)(7,8);(10,13)(14,16)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^b=fh,g^b=gh,e^c=eh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72226680, #72233910, #72238180, … |
| maximal normal subgroup of | #72226680, #72233910, #72238180, … |
| maximal quotient of | #72219603, #72230994, #72231276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92107);