Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511785 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #24222 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1190 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1190 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,19)(18,20);(1,2,3,4)(7,8)(11,19)(12,15)(16,17);(5,7,10,8)(6,11,14,19)(9,16,18,17)(12,13,15,20);(5,8,10,7)(6,12,14,15)(9,16,18,17)(11,13,19,20)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,d^b=di,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72312247, #72312263, #72312321, … |
| maximal normal subgroup of | #72312247, #72312263, #72312321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419112);