Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129565 |
| 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 | #3061 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 597 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,3,8,6)(9,10,15,20,16,12,14,19)(11,21,23,39,25,40,24,35)(13,28,31,27,32,36,30,18)(17,33,34,22,26,29,37,38);(1,3)(2,5)(4,6)(7,8)(9,11)(10,17)(12,26)(13,29)(14,23)(15,24)(16,25)(18,35)(19,34)(20,37)(21,36)(22,31)(27,39)(28,40)(30,38)(32,33);(2,3)(6,7)(9,12,16,10)(11,17,25,26)(13,27,32,18)(14,19,15,20)(21,38,40,22)(23,37,24,34)(28,30,36,31)(29,35,33,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dgi,f^b=fi,g^b=gi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36762);