Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1445 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20)(12,25,29,35)(13,30)(14,18)(15,19)(16,22)(17,28,36,27)(21,37,39,31)(23,33)(24,32)(26,40,38,34);(9,11)(10,16,19,33)(12,26)(13,22)(14,23)(15,24)(17,34,36,40)(18,20,30,32)(21,27)(25,37,35,31)(28,39)(29,38);(9,12)(10,17,19,36)(11,21)(13,27)(14,28)(15,29)(16,31,33,37)(18,25,30,35)(20,34,32,40)(22,38)(23,26)(24,39)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28949);