Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485617 |
| 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 | #18152 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 848 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:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,26)(14,18)(15,22)(17,28)(20,34)(21,35)(23,37)(24,39)(25,33)(27,29)(30,36)(31,38)(32,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,23)(13,27)(14,20)(16,31)(17,34)(18,29)(19,26)(21,25)(22,28)(24,36)(30,33)(32,35)(37,39)(38,40);(1,3,5,4)(2,6,7,8)(11,20)(13,28)(15,29)(19,34)(21,23)(24,40)(30,31)(35,37);(2,7)(6,8)(9,12,14,25)(10,16,18,33)(11,21,20,23)(13,24,28,40)(15,30,29,31)(17,32,26,39)(19,35,34,37)(22,36,27,38)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,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, 392967);