Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139956 |
| 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 | |
516 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 516 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:
(1,2,3,11,5,7,15,9)(4,17,19,28,20,10,30,25)(6,12,16,31,13,23,32,29)(8,24,27,18,22,21,26,14);(1,3,5,15)(2,7)(4,14,20,18)(6,16,13,32)(8,25)(9,11)(10,26)(12,23)(17,27)(19,24,30,21)(22,28)(29,31);(1,3,13,32)(2,8,23,25)(4,18,24,19)(5,15,6,16)(7,22,12,28)(9,26,31,10)(11,27,29,17)(14,21,30,20)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72233825, #72233835, #72233836, … |
| maximal normal subgroup of | #72233825, #72233835, #72233836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47241);