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 | #23900 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 1620 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,24)(12,15,20,23)(14,21,22,18);(2,5)(6,8)(12,20)(15,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24);(3,7)(6,8)Polycyclic presentation:
2^9|a^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 | #72231754, #72233080, #72233163, … |
| maximal normal subgroup of | #72231754, #72233080, #72233163, … |
| maximal quotient of | #72234963, C2×#508063, #72261418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415259);