Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186842 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62832 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2694 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 2694 conjugacy classes of subgroups — too many to draw. Only its 310 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:
(2,7,10,21)(3,12)(4,16)(8,23,26,20)(9,17,11,22)(13,15)(14,29)(18,31)(24,30,25,32)(27,28);(1,2)(3,8)(4,17)(5,11)(6,9)(7,18)(10,19)(12,26)(13,24)(14,30)(15,25)(16,21)(20,29)(22,31)(23,28)(27,32);(3,13)(4,18)(7,17)(8,24)(12,15)(14,27)(16,31)(20,23)(21,22)(25,26)(28,29)(30,32);(1,3,5,13)(2,8,11,24)(4,14,16,28)(6,15,19,12)(7,20,22,32)(9,25,10,26)(17,30,21,23)(18,29,31,27)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72232960, #72233035, #72233173, … |
| maximal normal subgroup of | #72232960, #72233035, #72233173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94039);