Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 | #61393 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1708 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)(4,8)(5,7)(9,10,14,18,15,19,13,27)(11,20,23,16,21,31,28,29)(12,17,26,33,24,32,25,37)(22,34,36,30,35,40,38,39);(1,3,6,2)(4,8,5,7)(11,21)(12,24)(16,29)(17,32)(20,31)(23,28)(25,26)(33,37);(1,4,6,5)(2,7,3,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,21)(17,30)(18,31)(19,29)(20,27)(24,35)(25,38)(26,36)(32,39)(33,40)(34,37);(1,4,6,5)(2,8,3,7);(1,5,6,4)(2,8,3,7)(9,12,14,26,15,24,13,25)(10,17,18,33,19,32,27,37)(11,22,23,36,21,35,28,38)(16,30,31,40,29,39,20,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=f,e^2=fh,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^b=ei,g^c=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72313312, #72315597, #72315607 |
| maximal normal subgroup of | #72313312, #72315597, #72315607 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6308185);