Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130482 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | #2397 | C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×D4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
867 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 867 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10)(11,19,22,32)(12,23,25,15)(13,26,28,17)(14,18)(16,21)(20,34)(24,30)(27,31)(29,35,37,39)(33,36)(38,40);(2,6)(3,4)(7,8)(10,15)(16,29)(17,30)(18,23)(19,33)(26,38)(31,37)(32,34);(9,11,14,22)(10,16,18,31)(12,20,25,36)(13,27,28,21)(15,29,23,37)(17,32,26,19)(24,39,40,35)(30,34,38,33)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72292982, #72292986, #72293175, … |
| maximal normal subgroup of | #72292982, #72292986, #72293175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37945);