Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #52943 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 602 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(2,5,6,7)(9,10,14,20,15,18,13,19)(11,21,23,36,24,16,29,32)(12,25,27,17,28,34,30,35)(22,37,39,31,26,40,38,33);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,26)(13,29)(14,23)(15,24)(17,33)(18,21)(19,32)(20,36)(22,28)(25,37)(27,38)(30,39)(31,35)(34,40);(3,8)(5,7)(9,12)(10,17)(11,22)(13,27)(14,30)(15,28)(16,31)(18,35)(19,25)(20,34)(21,33)(23,38)(24,26)(29,39)(32,40)(36,37)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,f^c=fgi,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39723);