Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476095 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2679 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
811 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 811 conjugacy classes of subgroups — too many to draw. Only its 186 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:
(1,2,6,8)(3,4,5,7)(9,10)(11,14)(12,20)(13,17)(15,18)(16,22)(19,24)(21,23);(1,3,6,5)(2,7,8,4)(9,11)(10,14)(12,21)(13,19)(15,16)(17,24)(18,22)(20,23);(1,4,6,7)(2,3,8,5)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,23)(18,22);(1,5,6,3)(2,4,8,7)(9,10)(11,15)(12,20)(13,17)(14,21)(16,22)(18,24)(19,23)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| derived subgroup of | #72227422, #72227428, #72227442 |
| Fitting subgroup of | #72227422 |
| maximal subgroup of | #72227422, #72227423 |
| maximal normal subgroup of | #72227422, #72227423 |
| maximal quotient of | #72268630 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383408);