Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139954 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
570 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 32 points:
(1,2,3,11,6,13,4,9)(5,12,21,31,19,32,20,28)(7,24,16,26,8,14,17,30)(10,23,25,15,27,22,29,18);(2,9,13,11)(5,19)(10,25,27,29)(12,31,32,28)(14,26,24,30)(15,18)(20,21)(22,23);(1,3,8,17)(2,10,14,28)(4,7,16,6)(5,18,23,20)(9,25,26,12)(11,29,30,32)(13,27,24,31)(15,22,21,19)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,e^b=ehi,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72262544, #72262738, #72262743 |
| maximal normal subgroup of | #72262544, #72262738, #72262743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47215);