Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485320 |
| 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 | #18350 | 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 | |
855 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 855 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)(12,23)(13,17)(14,18)(15,22)(16,25)(20,34)(21,33)(24,40)(26,30)(27,31)(28,38)(29,37)(32,35)(36,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,28)(17,19)(18,29)(23,39)(24,36)(25,35)(26,38)(27,37)(30,32)(31,33)(34,40);(9,12)(10,16,18,23)(11,20)(13,26)(14,25)(15,28,29,39)(17,30,31,40)(19,32,33,34)(21,35)(22,38)(24,27)(36,37);(1,3,5,4)(2,6,7,8)(11,21)(15,29)(19,33)(20,35)(22,37)(28,39)(32,34)(36,38)Polycyclic presentation:
2^9|c^2=hi,d^2=g,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,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392693);