Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #120636 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1442 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
674 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 674 conjugacy classes of subgroups — too many to draw. Only its 122 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)(3,6,8,4)(5,7)(9,10)(11,19,22,15)(12,23)(13,18)(14,17)(16,25)(20,29,37,35)(21,36,39,30)(24,31,38,40)(26,34)(27,32)(28,33);(1,3)(2,6,7,4)(5,8)(9,11)(10,15)(12,24,25,38)(13,21)(14,22)(16,31,23,40)(17,19)(18,30)(20,26,37,27)(28,39)(29,32,35,34)(33,36);(1,4,5,6)(2,3)(7,8)(9,12,14,25)(10,16)(11,20,22,37)(13,27,28,26)(15,29)(17,23)(18,34)(19,35)(21,38,39,24)(30,40)(31,36)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=dh,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fi,g^b=gh,e^c=eh,f^c=fh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27833);