Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508074 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #3486 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
990 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 990 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10)(3,12,14,8)(4,16,18,30)(6,15,22,31)(7,11,23,24)(9,26,27,17)(13,20,29,21)(19,32,28,25);(1,3,5,14)(2,8,10,12)(4,17,18,26)(6,24,22,11)(7,15,23,31)(9,16,27,30)(13,28,29,19)(20,32,21,25);(1,4)(2,9)(3,13)(5,18)(6,25)(7,21)(8,17)(10,27)(11,19)(12,26)(14,29)(15,16)(20,23)(22,32)(24,28)(30,31);(4,18)(9,27)(13,29)(16,30)(17,26)(19,28)(20,21)(25,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268097 |
| maximal subgroup of | #72268087, #72268097, #72268326, … |
| maximal normal subgroup of | #72268087, #72268097, #72268326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415342);