Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514060 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,3)(6,7)(9,10,13,12)(11,14,16,15);(1,2,4,3)(6,7)(9,11)(10,14)(12,15)(13,16);(5,6,8,7);(5,7,8,6)(10,12)(11,15)(14,16);(1,3,4,2)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72281173, #72281251, #72281491, … |
| maximal normal subgroup of | #72281173, #72281251, #72281491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470280);