Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1791 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1791 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,7)(4,6)(5,8)(12,19)(17,23);(1,3,4,8)(2,5,6,7)(10,14)(12,19)(15,21)(16,20)(17,23)(22,24);(9,10)(11,15)(12,20)(13,16)(14,19)(17,24)(18,22)(21,23);(2,6)(5,7)(9,11,13,18)(10,15,16,22)(12,17,19,23)(14,21,20,24)Polycyclic presentation:
2^9|b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72209805, #72209806, #72213210, … |
| maximal normal subgroup of | #72209805, #72213210, #72213317, … |
| maximal quotient of | #72227645, #72233153, #72236957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407206);