Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498890 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1621 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1621 conjugacy classes of subgroups — too many to draw. Only its 178 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:
(1,2,4,6)(3,7,8,5)(10,14)(11,17)(15,22)(16,21)(18,24)(20,23);(1,3,4,8)(2,5,6,7)(11,17)(12,19)(15,23)(20,22);(9,10)(11,15)(12,14)(13,16)(17,23)(18,20)(19,21)(22,24);(9,11,13,17)(10,15,16,23)(12,18,19,24)(14,20,21,22)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72255931, #72256210 |
| maximal normal subgroup of | #72255931, #72256210 |
| maximal quotient of | #72263332 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406181);