Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508062 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1232 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 128 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)(9,10,14,11)(12,19,21,24)(13,16,22,18)(15,23,17,20);(2,5)(6,8)(9,11,14,10)(12,17,21,15)(13,16,22,18)(19,20,24,23);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(3,7)(6,8)Polycyclic presentation:
2^9|a^2=i,b^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,f^b=fh,d^c=de,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72261306, #72303931 |
| maximal normal subgroup of | #72261306, #72303931 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415260);