Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(19,25)(22,28)(26,31)(29,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,d^c=di140 conjugacy classes — view the full character table.
| maximal subgroup of | #72287737, #72287744, #72287747, … |
| maximal normal subgroup of | #72287737, #72287744, #72287747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251567);