Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 | #60915 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#424 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,20,13,17)(11,16,23,30,24,31,22,28)(12,18,27,34,25,33,26,32)(21,29,37,40,35,39,36,38);(2,5)(3,7)(9,11,14,23,15,24,13,22)(10,16,19,30,20,31,17,28)(12,21,27,37,25,35,26,36)(18,29,34,40,33,39,32,38);(10,17,20,19)(16,28,31,30)(18,32,33,34)(29,38,39,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,26)(14,27)(15,25)(16,29)(17,32)(19,34)(20,33)(22,36)(23,37)(24,35)(28,38)(30,40)(31,39);(2,5)(6,8)(12,25)(18,33)(21,35)(26,27)(29,39)(32,34)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303515);