Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C2×H(5,𝔽2) | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16);(3,10)(7,14)(11,16)(13,15)Polycyclic presentation:
2^9|a^2=f,e^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,e^b=eg,d^c=dg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72320682, #72322022, #72322071 |
| maximal normal subgroup of | #72320682, #72322022, #72322071 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388412);