Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139538 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | #3497 | C22 |
| 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 | |
525 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,6)(7,8)(9,10,13,20,15,21,12,18)(11,22,23,16,25,33,26,34)(14,19,27,37,30,36,28,17)(24,31,40,38,29,35,39,32);(1,2,6,8,4,7,3,5)(9,11)(10,16,21,34)(12,26)(13,23)(14,29)(15,25)(17,35,37,31)(18,22,20,33)(19,32,36,38)(24,30)(27,39)(28,40);(9,12,15,13)(10,17,21,37)(11,23,25,26)(14,28,30,27)(16,31,34,35)(18,19,20,36)(22,38,33,32)(24,40,29,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,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=ci,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46751);