Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2315 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
573 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 573 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,19)(12,23)(13,27,28,17)(14,18)(15,21,30,37)(16,26)(20,35)(22,34)(24,33)(25,32,40,38)(29,39,31,36);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,18,30)(12,24,26,20)(13,21,28,37)(16,31,23,29)(17,19,27,34)(25,39,40,36)(32,35,38,33);(3,8)(5,7)(9,12)(10,16,18,23)(11,20)(13,25)(14,26)(15,29,30,31)(17,32,27,38)(19,33,34,35)(21,36)(22,24)(28,40)(37,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39221);