Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 98 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)(9,10,12,16,14,19,26,18)(11,20,23,37,24,38,21,35)(13,27,25,17,29,34,39,36)(15,30,32,22,33,28,31,40);(3,7)(5,8)(9,11,14,24)(10,15)(12,21,26,23)(13,28,29,30)(16,31)(17,35)(18,32)(19,33)(20,27)(22,39,40,25)(34,38)(36,37);(1,2,4,6)(3,8,7,5)(9,12,14,26)(10,16,19,18)(11,21,24,23)(13,25,29,39)(15,31,33,32)(17,34,36,27)(20,35,38,37)(22,30,40,28)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=hi,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309475, #72309481, #72309484, … |
| maximal normal subgroup of | #72309475, #72309481, #72309484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43575);