Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129559 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
624 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 108 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:
(1,2,4,7,5,3,8,6)(9,10)(11,20,22,16)(12,15)(13,27)(14,18)(17,24)(19,25)(21,33,38,28)(23,32,37,35)(26,29)(30,39)(31,36,34,40);(1,3)(2,5)(4,6)(7,8)(10,15)(12,24)(13,25)(16,28)(17,19)(18,29)(20,33)(21,36)(23,37)(26,39)(27,30)(31,32)(34,35)(38,40);(2,3)(6,7)(9,11)(10,16)(12,21)(13,23)(14,22)(15,28)(17,31)(18,20)(19,32)(24,36)(25,37)(26,38)(27,35)(29,33)(30,34)(39,40)Polycyclic presentation:
2^9|a^2=f,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dg,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36755);