Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1444 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
712 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 138 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)(9,10,14,18,15,19,13,29)(11,20,24,31,23,35,22,16)(12,17,28,32,27,33,26,39)(21,36,38,30,25,34,37,40);(1,2)(3,7,8,5)(4,6)(9,11,13,22,15,23,14,24)(10,16,18,20,19,31,29,35)(12,25,26,38,27,21,28,37)(17,30,32,34,33,40,39,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21,23,25)(13,26)(14,28)(15,27)(16,30,31,40)(18,32)(19,33)(20,34,35,36)(22,37,24,38)(29,39)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,d^c=di,f^c=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28561);