Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139602 |
| 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 | C2×C4×Q8 | C23 |
| prime radical | C23 | #386 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,12,18,15,20,13,27)(11,21,24,33,25,38,22,16)(14,19,28,17,30,35,26,36)(23,31,39,32,29,34,40,37);(1,2,3,5,4,6,7,8)(9,11)(10,16,20,33)(12,24)(13,22)(14,29)(15,25)(17,34,36,31)(18,21,27,38)(19,32,35,37)(23,30)(26,39)(28,40);(2,6)(5,8)(9,12,15,13)(10,17,20,36)(11,22,25,24)(14,28,30,26)(16,31,33,34)(18,19,27,35)(21,37,38,32)(23,40,29,39)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=cg,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46803);