Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130482 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#554 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
638 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 638 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6,7,8)(9,10,14,20,15,18,11,16)(12,23,21,36,24,39,29,38)(13,26,22,35,28,19,27,31)(17,32,33,40,34,37,30,25);(1,2)(3,6)(4,8)(5,7)(9,11,15,14)(10,16)(12,21,24,29)(13,27,28,22)(17,30)(18,20)(19,35)(23,38)(25,32,40,37)(26,31)(33,34)(36,39);(1,3,5,4)(2,6,7,8)(9,12,15,24)(10,17)(11,21,14,29)(13,25,28,40)(16,30)(18,34)(19,23)(20,33)(22,37,27,32)(26,39)(31,36)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=gh,d^2=hi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37900);