Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9467 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,7,5,8,3,6)(13,16);(2,6,8,7)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,15)(14,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^b=ei,d^c=di98 conjugacy classes — view the full character table.
| maximal subgroup of | #72231778, #72232797, #72234080, … |
| maximal normal subgroup of | #72231778, #72232797, #72234080, … |
| maximal quotient of | #72231763, #72234057, #72240194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15390);