Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119538 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
827 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 827 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2,4,11)(3,12,14,28)(5,7,16,21)(6,10,18,27)(8,22,24,29)(9,25,26,13)(15,23,30,32)(17,31,19,20);(2,7)(3,13)(4,16)(8,12)(9,23)(10,20)(14,22)(15,29)(17,18)(24,32)(25,30)(26,28);(2,8,7,9)(4,17)(10,23,20,12)(11,28,21,32)(14,22)(16,18)(24,27,26,31)(25,30)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gh,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72235262, #72237976, #72237989, … |
| maximal normal subgroup of | #72235262, #72237976, #72237989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26734);