Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10336 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1468 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1468 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,18,14,17,25,31)(11,15,21,27,20,28,33,38)(12,16,24,30,23,29,36,39)(19,26,32,37,22,34,35,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,22,23,19)(13,21)(14,20)(16,26,29,34)(17,28)(18,27)(24,35,36,32)(25,33)(30,37,39,40)(31,38);(5,8)(9,12)(10,16)(11,19)(13,24)(14,23)(15,26)(17,29)(18,30)(20,22)(21,32)(25,36)(27,37)(28,34)(31,39)(33,35)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=d,g^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23549);