Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| 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 | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61437 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,5,8,7)(9,10,13,17,14,11,15,19)(12,16,21,23,20,18,22,24);(2,5)(3,8)(6,7);(2,6)(5,7)(9,11,13,19,14,10,15,17)(12,18,21,24,20,16,22,23);(2,6)(5,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24);(12,20)(16,18)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=g,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dh,f^a=fh,f^b=fh,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72216884, #72287796, #72309616, … |
| maximal normal subgroup of | #72216884, #72287796, #72309616, … |
| maximal quotient of | #72225125, #72286485, #72286502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253017);