Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485060 |
| 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 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
962 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 962 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2)(3,5)(4,6)(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);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,32)(18,29)(22,39)(23,35)(24,36)(25,30)(26,38)(27,37)(31,40)(33,34);(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);(2,6)(5,7)(11,20)(15,29)(19,35)(21,37)(28,40)(30,32)(34,39)(36,38)Polycyclic presentation:
2^9|c^2=i,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,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, 392257);