Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23913 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1177 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1177 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,22)(19,23)(21,24);(2,5)(6,8)(9,11,13,18)(10,14,17,22)(12,19,20,24)(15,23,16,21);(1,3,4,7)(2,6,5,8)(12,20)(19,24);(10,15)(11,18)(12,20)(14,21)(16,17)(22,23)Polycyclic presentation:
2^9|b^2=i,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,d^b=di,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285788, #72285929, #72286113, … |
| maximal normal subgroup of | #72285788, #72285929, #72286113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419333);