Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153537 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13584 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #494 | C23 |
| prime radical | #19750 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3152 ◁ G | |
| LECS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,4,10,7,12,20,27)(3,9,15,25,14,24,29,32)(5,11,19,28,6,21,18,26)(8,17,23,16,22,30,31,13);(2,8)(3,13)(5,6)(9,11)(10,25)(12,22)(14,16)(15,17)(18,19)(21,24)(23,26)(27,32)(28,31)(29,30);(3,14)(8,22)(9,24)(13,16)(15,29)(17,30)(23,31)(25,32)Polycyclic presentation:
2^9|a^2=e,d^2=gh,e^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dghi,e^b=efg,f^b=fhi,g^b=gi,d^c=di,e^d=egh,f^e=fhi,g^e=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72340153, #72340156, #72340161, … |
| maximal normal subgroup of | #72340153, #72340156, #72340161, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60737);