Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
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 | C23×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,14,18,15,19,13,28)(11,20,23,30,24,34,22,16)(12,17,27,32,25,31,26,38)(21,33,37,39,35,40,36,29);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8)(9,11,13,22,15,24,14,23)(10,16,18,20,19,30,28,34)(12,21,26,36,25,35,27,37)(17,29,32,33,31,39,38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(22,36)(23,37)(24,35)(28,38)(30,39)(34,40);(12,25)(17,31)(21,35)(26,27)(29,39)(32,38)(33,40)(36,37)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282208);