Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487635 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #25773 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
935 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 935 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,25)(14,17)(15,22)(18,28)(20,34)(21,33)(23,32)(24,38)(26,30)(27,31)(29,35)(36,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,17,30)(12,20,23,36)(13,26,28,22)(16,29,32,39)(18,33,25,19)(24,37,27,35)(31,40,38,34);(3,4)(6,8)(9,12)(10,16)(11,20)(13,27)(14,23)(15,29)(17,32)(18,31)(19,34)(21,36)(22,37)(24,28)(25,38)(26,35)(30,39)(33,40);(1,3,5,4)(2,6,7,8)(10,17)(11,21)(12,23)(18,25)(22,26)(24,27)(29,39)(34,40)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395100);