Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #412 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,3)(2,6)(4,8)(5,7);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,f^b=fh,d^c=dhi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72321691, #72322064, #72322149 |
| maximal normal subgroup of | #72321691, #72322064, #72322149 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250258);