Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132529 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 556 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)(11,21)(12,25,29,36)(13,19,14,18)(15,20)(16,23,33,30)(17,28)(22,34,26,31)(24,32)(27,35)(37,38)(39,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16,20,33)(12,26)(13,30)(14,23)(15,24)(17,34,35,31)(18,32,19,21)(22,29)(25,37,36,40)(27,38)(28,39);(3,8)(5,7)(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,31)(18,25)(19,36)(20,35)(21,37)(23,39)(24,26)(30,38)(32,40)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fghi,d^c=dgh,e^c=ei,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, 39871);