Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | #1790 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
675 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 675 conjugacy classes of subgroups — too many to draw. Only its 114 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)(11,19,22,34)(12,16)(13,26,28,17)(14,18)(15,21)(20,37,23,36)(24,35,40,32)(25,33)(27,30)(29,38)(31,39);(9,11,14,22)(10,15,18,30)(12,23,25,20)(13,27,28,21)(16,31,33,29)(17,34,26,19)(24,38,40,39)(32,37,35,36);(9,12,14,25)(10,16,18,33)(11,20,22,23)(13,24,28,40)(15,29,30,31)(17,35,26,32)(19,36,34,37)(21,38,27,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,e^b=egi,f^b=fgi,d^c=dg,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282351, #72282398, #72282516, … |
| maximal normal subgroup of | #72282351, #72282398, #72282516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36808);