Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 | #61401 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10,13,17,14,18,22,16)(11,19,12,15,20,24,21,23);(1,3,6,2)(4,7,5,8)(11,20)(12,21)(15,23)(19,24);(9,11,13,12,14,20,22,21)(10,15,16,19,18,23,17,24);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,7,3,8)(9,12,13,20,14,21,22,11)(10,15,17,24,18,23,16,19)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,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,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72313983, #72315597, #72315643 |
| maximal normal subgroup of | #72313983, #72315597, #72315643 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311816);