Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487699 |
| 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×C4⋊C4 | C24 |
| prime radical | #18350 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 871 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)(11,15)(12,22)(13,25)(14,18)(16,24)(17,27)(19,34)(20,29)(21,32)(23,33)(26,31)(28,36)(30,37)(35,39)(38,40);(1,2,5,7)(3,6,4,8)(9,11,14,20)(10,15,18,29)(12,19,23,35)(13,21,27,37)(16,28,31,40)(17,32,25,30)(22,39,33,34)(24,36,26,38);(3,4)(6,8)(9,12,14,23)(10,16,18,31)(11,19,20,35)(13,26,27,24)(15,28,29,40)(17,33,25,22)(21,38,37,36)(30,39,32,34);(1,3,5,4)(2,6,7,8)(11,20)(15,29)(19,35)(21,37)(28,40)(30,32)(34,39)(36,38)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395298);