Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501084 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1926 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) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(5,6)(7,10,12,9)(8,11);(1,2)(3,4)(6,9)(7,12)(10,11);(2,4)(7,12);(6,9)(7,12)(10,11)Polycyclic presentation:
2^9|a^2=f,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,e^c=eh,e^d=ei,f^d=fhi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72234129, D4×#2196, #72234132, … |
| maximal normal subgroup of | #72234129, D4×#2196, #72234132, … |
| maximal quotient of | #72234160, #72234169, #72284398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408280);