Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,20,29,18)(11,16,23,33,24,34,39,32)(12,25,13,17,27,35,28,36)(21,37,38,30,26,40,22,31);(2,5)(3,7)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,31)(18,32)(19,33)(20,34)(21,27)(25,40)(28,38)(29,39)(30,36)(35,37);(9,12,14,13,15,27,29,28)(10,17,18,25,20,36,19,35)(11,21,23,38,24,26,39,22)(16,30,32,37,34,31,33,40);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(9,13,14,27,15,28,29,12)(10,17,19,35,20,36,18,25)(11,22,23,21,24,38,39,26)(16,31,33,37,34,30,32,40)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311244);