Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501156 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
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 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1971 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1971 conjugacy classes of subgroups — too many to draw. Only its 150 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:
(2,3)(6,9)(10,12);(1,2,4,3)(7,11);(5,6,7,9)(8,10,11,12);(1,3,4,2)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2196, #72234133, #72236574, … |
| maximal normal subgroup of | D4×#2196, #72234133, #72236574, … |
| maximal quotient of | #72234156, #72318221, #72318289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410378);