Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511775 |
| 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 | #23906 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1480 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1480 conjugacy classes of subgroups — too many to draw. Only its 144 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)(11,14)(12,15)(13,17)(16,21)(18,23)(19,24)(20,22);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,20)(13,18)(15,24)(16,22)(17,23)(19,21);(10,15)(11,18)(12,21)(14,22)(16,17)(23,24)Polycyclic presentation:
2^9|c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,d^c=di,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72210575, #72215568, #72216603, … |
| maximal normal subgroup of | #72210575, #72215568, #72216603, … |
| maximal quotient of | #72216584, #72219521, C2×#512119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419315);