Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,3,5,4,6,7,8)(9,10,11,17,14,20,15,31)(12,23,21,34,26,16,27,32)(13,28,22,36,30,37,29,19)(18,35,33,24,38,40,39,25);(10,16)(11,15)(12,24)(13,29)(17,34)(18,36)(19,38)(20,23)(21,35)(22,30)(25,26)(27,40)(28,33)(31,32)(37,39);(9,11,14,15)(10,17,20,31)(12,21,26,27)(13,22,30,29)(16,32,23,34)(18,33,38,39)(19,28,36,37)(24,40,25,35)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,e^b=ei,f^b=fh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72223651, #72223673, C8×#1419, … |
| maximal normal subgroup of | #72223651, #72223673, C8×#1419, … |
| maximal quotient of | C2×#150064, #72226999, #72286707 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57260);